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Question 1159323: A picture frame holds an 8-inch by 10-inch photograph. The frame adds a border x inches wide around three sides of the photo. On the fourth side, the frame is wider to accommodate a decoration on the frame. The forth side is (3x-1) inches wide, as shown in the figure.
a) write a quadratic expression for the combined area of the frame in terms of x.
b)if the border on the three matching sides in 1 inch, what is the combined area of the frame?
c)if the combined area of the frame is 165 square inches, find x.
Answer by ikleyn(52784)

(Show Source):

You can put this solution on YOUR website!
.
A picture frame holds an 8-inch by 10-inch photograph. The frame adds a border x inches wide around three sides
of the photo. On the highlight%28cross%28forth%29%29

fourth side, the frame is wider to accommodate a decoration on the frame.
The highlight%28cross%28forth%29%29

fourth side is (3x-1) inches wide, as shown in the figure.
(a) write a quadratic expression for the combined area of the frame in terms of x.
(b) if the border on the three matching sides in 1 inch, what is the combined area of the frame?
(c) if the combined area of the frame is 165 square inches, find x.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a)  The Figure is not shown, so the reference to the Figure is empty (carries zero information).

     @MowMow assumes that the fourth side of (3x-1) inches wide is the 8 inches long.
     I will follow to this assumption in order to show that the "solution" by @MowMow is inadequate.

     Indeed, then the outer dimensions of the frame are (10+x+(3x-1)) = (9+4x) inches by (8+2x) inches,
     which is DIFFERENT from (10+3x) and (8+2x) inches, used in the post by @MowMow.

     Then the quadratic expression for the combined area of the frame should be different from that 
     in the post by @MowMow.


     As the request (a) is presented in the post, it baffles me.

     If they request for the area of the large rectangle with the sides (9+4x) and (8+2x),
     then the area is the product (9+4x)*(8+2x) square inches.

     If they want the area of the frame only, then this area is  (9+4x)*(8+2x) - 8*10 square inches.


     Actually, the request in (a) is written in such an uncertain way, that the precise meaning
     is unclear.  It is not a way to write a Math problem.



(b)  When I read the request (b), it baffles me again.

     Indeed, it says that the border of the three matching sides is 1 inch, but says nothing about the fourth side.

     In any case, the answer by @MowMow is irrelevant.



(c)  The same uncertainty is with the request (c).

     From their formulation, it is unclear, if we are under condition (a) or under condition (b).



As a final resume, my opinion is that 

    (a) this writer, who created the problem, is not able to generate a Math problem 
        by an non-ambiguous way, as it is required for Math problems.

    (b) this post is not an adequate Math problem and can not be considered as a correct Math problem.

    (c) For any interpretation, the solution and the answers by @MowMow are incorrect.


For a reader, my advice is to ignore this post as inaccurate and ambiguous 
and to ignore the "solution" by @MowMow as inadequate.

I know that all solutions and all posts are included into databases for artificial intelligence.

So I want to warn: DO NOT INCLUDE IT in the database for AI, in any form.

Simply re-direct it into the garbage bin, where its right place should be.

The author of this problem deserves a big minus and a serious warning
for unprofessionalism in writing Math problems.

Question 1165819: Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs, the cost of each chair, $y is given by y = x2 - 14x + 80.
a. Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
b. Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!

Here are alternative methods for solving each of the two parts of the problem.

The graph of the cost function is a parabola, with its minimum value when the number of chairs ordered is

a. Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.

Since the graph is symmetric about the axis of symmetry x=7, the cost will be the same for 9 chairs as it is for 5 chairs.

ANSWER: 9 chairs can be ordered for the same cost as 5 chairs

b. Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.

The minimum cost, when 7 chairs are ordered, is

Because the leading term of the cost function is , the cost will differ from the minimum cost of $31 by n^2 when x differs from 7 by n. We need the cost to be less than $45, which differs from the minimum cost of $31 by $14. Since 3^2 is less than 14 and 4^2 is greater than 14, the number of chairs that can be ordered for less than $45 can differ from 7 by at most 3.

ANSWER: Between 4 and 10 chairs can be ordered for a cost less than $45

Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs,
the cost of each chair, $y is given by y = x2 - 14x + 80.
(a) Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
(b) Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a) If he orders 5 chairs, the cost of each chair is 5^2 - 14*5 + 80 = 35 dollars. To find x, we should solve this quadratic equation x^2 - 14x + 80 = 35. This equation is the same as x^2 - 14x + 80 - 35 = 0, x^2 - 14x + 45 = 0, (x-5)*(x-9) = 0 (after factoring). One root is 5 chairs (the value we started with). The other root is 9, which is the ANSWER to question (a). (b) To answer (b), we should solve this inequality x^2 - 14x + 80 <= 45. Transform and simplify it x^2 - 14x + 80 - 45 <= 0, x^2 - 14x + 35 <= 0, Apply the quadratic formula to find the roots. The roots are = . So, one root is = 3.26 (approx.) and another root is = 10.74 (approx.) Function x^2 - 14x + 35 is negative between the roots. Since we are interested to know integer values of x, they are between 4 and 10 inclusive. So, the ANSWER to question (b) is "integer numbers between 4 and 10 inclusive".

Solved.

Question 1165818: Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs, the cost of each chair, $y is given by y = x^2 - 14x + 80.
a. Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
b. Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Mr. Lim orders chairs from a factory to sell. When he orders a batch of x chairs,
the cost of each chair, $y is given by y = x2 - 14x + 80.
(a) Find x such that the cost of each chair is the same as when Mr. Lim
ordered 5 chairs.
(b) Find the number of chairs in a batch he needs to orders such that the cost
per chair is less than $45.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a) If he orders 5 chairs, the cost of each chair is 5^2 - 14*5 + 80 = 35 dollars. To find x, we should solve this quadratic equation x^2 - 14x + 80 = 35. This equation is the same as x^2 - 14x + 80 - 35 = 0, x^2 - 14x + 45 = 0, (x-5)*(x-9) = 0 (after factoring). One root is 5 chairs (the value we started with). The other root is 8, which is the ANSWER to question (a). (b) To answer (b), we should solve this inequality x^2 - 14x + 80 <= 45. Transform and simplify it x^2 - 14x + 80 - 45 <= 0, x^2 - 14x + 35 <= 0, Apply the quadratic formula to find the roots. The roots are = . So, one root is = 3.26 (approx.) and another root is = 10.74 (approx.) Function x^2 - 14x + 35 is negative between the roots. Since we are interested to know integer values of x, they are between 4 and 10 inclusive. So, the ANSWER to question (b) is "integer numbers between 4 and 10 inclusive".

Solved.

Question 1210355: what does the 'a' in the quadratic equation y=ax^2+bx+c represent?

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

The 'a' is the leading coefficient.
a, b, and c are placeholders for values.

If a > 0 then the parabola opens upward.
For example consider y = 3x^2+5x-2 where a = 3

In contrast, now consider y = -4x^2-7x+1 where a = -4 this time.
The parabola opens downward because a < 0.

I recommend using either Desmos or GeoGebra as a graphing tool.
There are many other similar tools as well.

Here is an interactive Desmos graph where you can adjust the value of 'a' to any number you want.
Move the slider around to see how the graph changes.
https://www.desmos.com/calculator/e0boawqc3q

Question 1180642: 1. A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h = - 0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet. A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h = - 0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.
a.Create a own quadratic relation that would model the path of a shot from a distance of 15 feet that would hit the rim of the basketball net. Explain how you obtained the answer.
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h = - 0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet. A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h = - 0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.
a.Create a own quadratic relation that would model the path of a shot from a distance of 15 feet that would hit the rim of the basketball net. Explain how you obtained the answer.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This text in the post is not a formulation of a mathematical problem.

This is a collection of words without clear meaning behind it.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to create a quadratic relation that models a basketball shot hitting the rim from 15 feet away:
**Understanding the Problem:**
* We're given the standard quadratic model for a basketball shot: h = -0.2d² + 3d + 6
* h: height of the ball
* d: distance from the shooter
* We need to create a new quadratic model where:
* The shot is taken from 15 feet away (d = 15).
* The ball hits the rim (we'll assume a standard rim height of 10 feet, so h = 10 when d = 15).
**Creating the New Quadratic Model:**
1. **General Form:** Start with the general form of a quadratic relation: h = ad² + bd + c
2. **Use the Given Information:**
* We know the shot is from 15 feet and hits the 10-foot rim, so we have a point (d, h) = (15, 10). Substitute this into the equation:
10 = a(15)² + b(15) + c
10 = 225a + 15b + c
* To make the shot hit the rim at this distance, we need the vertex of the parabola to be at d = 15. The x-coordinate (in this case, d-coordinate) of the vertex of the parabola is given by -b/2a. So:
15 = -b / 2a
-30a = b
3. **Choose a Value for 'a':**
* We have some freedom here. Let's choose a value for 'a' that's different from the original model but still makes sense for a basketball shot. A slightly smaller value for 'a' would make the shot arc higher. Let's try a = -0.15.
4. **Solve for 'b' and 'c':**
* Using -30a = b and a = -0.15, we get:
b = -30 * (-0.15) = 4.5
* Substitute a = -0.15 and b = 4.5 into the equation 10 = 225a + 15b + c:
10 = 225(-0.15) + 15(4.5) + c
10 = -33.75 + 67.5 + c
c = -23.75
**The New Quadratic Model:**
* h = -0.15d² + 4.5d - 23.75
This model represents a basketball shot taken from 15 feet away that hits the rim of the basketball net.
**Explanation:**
* We used the given information (distance and height of the rim) to create an equation with the general form of a quadratic relation.
* We used the fact that the vertex of the parabola should be at the rim (d = 15) to relate 'a' and 'b'.
* By choosing a value for 'a', we could solve for 'b' and 'c' to complete the model.

Question 1210323: X^2-2x+1 = 0
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website!

........apply the rule

solution:

Question 1181712: A rocket is being launched vertically over a point 𝐴 on the ground with
a velocity of 550 𝑚𝑖Τℎ𝑟. Twenty five miles away from point 𝐴 on the
ground, there is a photographer video-taping the launch. At what rate
is the angle of elevation of the camera changing when the rocket
achieves an altitude of 25 miles?
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
A rocket is being launched vertically over a point 𝐴 on the ground with
a velocity of 550 𝑚𝑖/ℎ𝑟. Twenty five miles away from point 𝐴 on the
ground, there is a photographer video-taping the launch. At what rate
is the angle of elevation of the camera changing when the rocket
achieves an altitude of 25 miles?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @CPhill is incorrect and his answer is incorrect, too.

        The error is in the last step, where @CPhill converts the value of 11 mi/hr to radians/hr.

        @CPhill mistakenly treats this value of 11 mi/hr as miles per hour.

        Actually, it is 11 radians per hour, and conversion is just done on the way and IS NOT NEEDED anymore.

        Below is my correct solution.

Here's how to solve this related rates problem: **1. Diagram and Variables:** * Draw a right triangle. * Point A is one vertex (where the rocket launches). * The photographer is at another vertex, 25 miles away from A. * The rocket's altitude is the vertical leg of the triangle (let's call it *y*). * The distance from A to the photographer is the horizontal leg (25 miles). * The angle of elevation from the photographer to the rocket is θ. **2. Given Information:** * dy/dt = 550 mi/hr (rocket's velocity) * We want to find dθ/dt when y = 25 miles. **3. Relate Variables:** We can relate θ and y using the tangent function: tan(θ) = y / 25 **4. Implicit Differentiation:** Differentiate both sides of the equation with respect to time (t): sec²(θ) * (dθ/dt) = (1/25) * (dy/dt) **5. Solve for dθ/dt:** dθ/dt = (1/25) * (dy/dt) / sec²(θ) dθ/dt = (cos²(θ)/25) * (dy/dt) **6. Find cos(θ) when y = 25 miles:** When y = 25 miles, the triangle is a right isosceles triangle, so θ = 45 degrees or π/4 radians. Therefore, cos(θ) = cos(45°) = 1/√2. **7. Substitute and Calculate:** dθ/dt = ((1/√2)² / 25) * 550 mi/hr dθ/dt = (1/50) * 550 mi/hr = 11 radians/hr. +-----------------------------------------------------------------------------+ | Conversion from mi/hr of the left side to radians/hr in the right side | | is just made inside the previous formula. | | This conversion is already built into the coefficients. | +-----------------------------------------------------------------------------+ **Answer:** The angle of elevation is changing at a rate of 11 radians per hour when the rocket reaches an altitude of 25 miles.

Solved and answered correctly.

------------------------------

Post-solution note

        11 radiance per hour is = 10.50423 degrees per minute,

        if you want to have the answer in units that are adequate to the problem.

///////////////////////////////////////////////////////

This problem was posted to this forum several years ago (about 5 years ago).

It was solved by Edwin under this link

https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1181710.html

https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1181710.html

Edwin' solution was conceptually correct, but had an error in his implementation, leading to incorrect answer.

I found that solution via Google search, and placed there my corrected solution.
My solution follows to the Edwin's idea/design, but fixes/repairs that error.

Now both my solutions to this problem produce the same answer, bringing peace in your mind.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this related rates problem:
**1. Diagram and Variables:**
* Draw a right triangle.
* Point A is one vertex (where the rocket launches).
* The photographer is at another vertex, 25 miles away from A.
* The rocket's altitude is the vertical leg of the triangle (let's call it *y*).
* The distance from A to the photographer is the horizontal leg (25 miles).
* The angle of elevation from the photographer to the rocket is θ.
**2. Given Information:**
* dy/dt = 550 mi/hr (rocket's velocity)
* We want to find dθ/dt when y = 25 miles.
**3. Relate Variables:**
We can relate θ and y using the tangent function:
tan(θ) = y / 25
**4. Implicit Differentiation:**
Differentiate both sides of the equation with respect to time (t):
sec²(θ) * (dθ/dt) = (1/25) * (dy/dt)
**5. Solve for dθ/dt:**
dθ/dt = (1/25) * (dy/dt) / sec²(θ)
dθ/dt = (cos²(θ)/25) * (dy/dt)
**6. Find cos(θ) when y = 25 miles:**
When y = 25 miles, the triangle is a right isosceles triangle, so θ = 45 degrees or π/4 radians. Therefore, cos(θ) = cos(45°) = 1/√2.
**7. Substitute and Calculate:**
dθ/dt = ((1/√2)² / 25) * 550 mi/hr
dθ/dt = (1/50) * 550 mi/hr = 11 mi/hr
**8. Convert to radians per hour:**
Since angular velocity is typically expressed in radians per unit time, we need to convert miles/hour to radians/hour. The relationship between arc length (s), radius (r), and angle (θ in radians) is s = rθ. If we consider the distance from the observer to the launch point as the radius (25 miles), and the altitude of the rocket as the arc length (25 miles), then when the altitude is 25 miles, the angle of elevation is π/4 radians. The rate of change of the rocket's altitude is given in miles per hour. The rate of change of the angle of elevation will be in radians per hour.
When y = 25 miles, the angle of elevation is 45 degrees or π/4 radians.
dθ/dt = 11 miles/hour * (1 radian / 25 miles) = 0.44 radians/hour.
**Answer:** The angle of elevation is changing at a rate of 0.44 radians per hour when the rocket reaches an altitude of 25 miles.

Question 1210275: Number that multiplies to -44 but adds to -1

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.

Here is another solution.

Since the sum of the numbers is -1, it means that the numbers are equally remoted from -0.5, so we can write a = -0.5 + x, b = -0.5 - x. Thus a + b = -1 is provided, and we should satisfy the other condition a*b = -44. It gives (-0.5+x)*(-0.5-x) = -44, - = -44, 0.025 + 44 = = 44.25 x = . Thus, the numbers are a = = , b = = .

Solved.

-------------------------

I posted this solution to surprise you that such simple problem may have two different approaches.
My other goal was to show you a solution which has a geometric meaning - not only a calculation exercise.

Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Numbers that multiply to -44 but add to -1.

Trial and error, or any other method, shows that there are not any two rational numbers that satisfy the conditions. So...

-->

Solve using the quadratic formula.

ANSWER: the two numbers are and

Question 1210274: Maricopa's Success scholarship fund receives a gift of $ 100000. The money is invested in stocks, bonds, and CDs. CDs pay 4.75 % interest, bonds pay 5.5 % interest, and stocks pay 6.7 % interest. Maricopa Success invests $ 40000 more in bonds than in CDs. If the annual income from the investments is $ 5590 , how much was invested in each account?
Found 3 solutions by mccravyedwin, ikleyn, Edwin McCravy:
Answer by mccravyedwin(407)   (Show Source):
You can put this solution on YOUR website!

But the most efficient way is to translate each of the three sentences into equations verbatim and use a solver. These days, humans do the translating and technology does the manipulating. Math has changed. Edwin

Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Maricopa's Success scholarship fund receives a gift of $100,000.
The money is invested in stocks, bonds, and CDs.
CDs pay 4.75 % interest, bonds pay 5.5 % interest, and stocks pay 6.7 % interest.
Maricopa Success invests $40,000 more in bonds than in CDs.
If the annual income from the investments is $5,590 , how much was invested in each account?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This problem is very special. It asks about three unknowns, but can be easily reduced
        to one single equation in one unknown.

        So, to get the finish and the answer, you don't need to write and to solve system of three
        linear equations. Respectively, using this approach, you don't need to know the methods
        of solving systems of equation.

Let x be the amount invested in CDs, in dollars. Then the amount invested in bonds is (x+40,000) dollars. Then the amount invested in stocks is (100,000 - x - (x+40,000)) = (60000-2x). Now, write this equation for the total annual interest 0.0475x + 0.055*(x+40000) + 0.067*(60000-2x) = 5590 dollars. Simplify it step by step and find x 0.0475x + 0.055x + 2200 + 4020 - 0.134x = 5590, 0.0475x + 0.055x - 0.134x = 5590 - 2200 - 4020, -0.0315x = -630 x = = 20000. So, $20,000 were invested in CDs; $20,000 + $40,000 = $60,000 were invested in bonds; and the rest, #100,000 - $20,000 - 60,000 = $20,000 were invested in stocks.

At this point, the problem is solved completely using one equation in one unknown.

In my solution, I even did not pronounce these words " system of equations ".

---------------------------

This approach teaches students to solve such problems in 3 (three) unknowns in early age.

So, it develops their mind, which is a major goal of mathematical education.

Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!

Solve the system and get B=$60000, C = $20000, and S=$20000. Here's an online solver to solve it for you so you don't have to do anything: https://www.emathhelp.net/calculators/algebra-2/system-of-equations-solver/ Edwin

Question 1165727: a satellite dish has a shaped of a paraboloid. If the receiver of the satellite dish is placed at the focus 2.53 ft from the vertex, write an equation for the cross-section of the satellite dish. Assume that the focus is on the positive x-axis and its vertex at the origin.
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
a satellite dish has a shape of a paraboloid. If the receiver of the satellite dish
is placed at the focus 2.53 ft from the vertex, write an equation for the cross-section
of the satellite dish. Assume that the focus is on the positive x-axis and its vertex at the origin.
~~~~~~~~~~~~~~~~~~~~~

        It is a standard problem of this kind. The feature is
        that in this problem the symmetry line is parallel to x-axis,
        while usually in such problems the symmetry line is parallel to y-axis.
        So, I will adapt a standard solution to this case.

For solving such problems, write an equation of the parabola in the cross-section in the form x = . (1) The advantage of writing in this form is the fact that then "p" is the distance from the parabola vertex to its focus. In this problem, the value of p is given: it is 2.53 ft. So, we substitute this value into equation (1), and we get x = . It gives the equation of the parabolic section x = , or x = . ANSWER

Solved.

Question 1210152: How to solve equation
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!

not factorable.

(this part removed)

Trying general formula solution directly:

Question 1170219: When lithium fluoride is dissolved in water, the equation of the solubility constant (Ksp) is
Ksp = [Li+] × [F-].
The value of Ksp for lithium fluoride is 0.00184 mol2/L2.
If the concentration of fluorine, [F-], is 0.0060 mol/L more that the concentration of lithium, [Li+], what is the concentration of lithium, [Li+], in the solution?
Round your answer to 2 significant figures.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step by step.
**1. Set Up the Equations**
* We are given the solubility constant (Ksp) expression: Ksp = [Li+] × [F-]
* We are given the value of Ksp: Ksp = 0.00184 mol²/L²
* We are given the relationship between [F-] and [Li+]: [F-] = [Li+] + 0.0060 mol/L
**2. Substitute and Solve**
* Substitute the expression for [F-] into the Ksp equation:
0.00184 = [Li+] × ([Li+] + 0.0060)
* Expand the equation:
0.00184 = [Li+]² + 0.0060[Li+]
* Rearrange the equation into a quadratic equation:
[Li+]² + 0.0060[Li+] - 0.00184 = 0
**3. Use the Quadratic Formula**
* The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a
* In our equation, a = 1, b = 0.0060, and c = -0.00184
* Substitute these values into the quadratic formula:
[Li+] = (-0.0060 ± √((0.0060)² - 4 × 1 × (-0.00184))) / (2 × 1)
[Li+] = (-0.0060 ± √(0.000036 + 0.00736)) / 2
[Li+] = (-0.0060 ± √0.007396) / 2
[Li+] = (-0.0060 ± 0.0860) / 2
**4. Find the Two Possible Solutions**
* Solution 1: [Li+] = (-0.0060 + 0.0860) / 2 = 0.0800 / 2 = 0.0400 mol/L
* Solution 2: [Li+] = (-0.0060 - 0.0860) / 2 = -0.0920 / 2 = -0.0460 mol/L
**5. Choose the Valid Solution**
* Since concentration cannot be negative, we discard the second solution.
* Therefore, [Li+] = 0.0400 mol/L
**6. Round to 2 Significant Figures**
* 0.0400 rounded to 2 significant figures is 0.040 mol/L.
**Answer:**
The concentration of lithium, [Li+], in the solution is 0.040 mol/L.

Question 1170322: Formulate a quadratic equation then answer what is asked in the problem.
John can answer all the problem solving in the activity in Math 9 module within 24 hours alone. But Justine can answer it 4 hours more than John. How long will it take Justine to answer? How many hours to finish answering the activity if they worked together?
Found 2 solutions by ikleyn, josgarithmetic:
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Formulate a quadratic equation then answer what is asked in the problem.
John can answer all the problem solving in the activity in Math 9 module within 24 hours alone.
But Justine can answer it 4 hours more than John. How long will it take Justine to answer?
How many hours to finish answering the activity if they worked together?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To solve this problem, quadratic equation is not needed.

Moreover, quadratic equation has no any relation to this problem.

                God, save us from such teachers,
                who themselves know nothing,
                but, nevertheless, compose nonsensical Math "problems"
                and disseminate them in the Internet.

Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!
Means, John needs 24 hours alone and Justine needs 28 hours alone.
(You say "within 24 hour" so maybe you need an inequality instead of equations.)
(I'm ignoring the "within" part.)

John

Justine

If both together, speed is , units of PROBLEM SET per HOURS.
Work with it!

Actually if question was only How long will it take Justine to answer, that was already given almost plainly. She needs no more than 28 hours.

Question 1209896: Let x and y be real numbers. Find the maximum value of (x + y)^2, if x and y satisfy x^2 + y^2 = 5 + 2xy.

Found 3 solutions by Edwin McCravy, ikleyn, CPhill:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!

Let x and y be real numbers. Find the maximum value of (x + y)^2, if x and y satisfy x^2 + y^2 = 5 + 2xy. Add 2xy to both sides of The left side is what we want to find a maximum value for. But! --- see the +4xy term on the right? You can choose x and y each to be a trillion trillion trillion trillion and even that would not be anywhere close to being a maximum value for . Answer: There is no maximum value for ! Edwin

Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Let x and y be real numbers. Find the maximum value of (x + y)^2,
if x and y satisfy x^2 + y^2 = 5 + 2xy.
~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @CPhill are INCORRECT.
        I write this my post to show / (to prove) that they are incorrect.

Equation x^2 + y^2 = 5 + 2xy is equivalent to x^2 - 2xy + y^2 = 5, (x-y)^2 = 5 x - y = +/- . It describes two parallel straight lines in a coordinate plane y = x + and y = x - . Moving the abscissa 'x' far enough to the right along x-axis in the positive domain, we can make value of 'x' as big as we like. Then the value of 'y' becomes big enough. Hence, the value of (x+y) can be made as big as we want. Then the value of (x+y)^2 becomes as big as we want and even bigger. So, expression (x+y)^2 DOES NOT HAVE MAXIMUM under the given restriction. ANSWER. Expression (x+y)^2 DOES NOT HAVE MAXIMUM under the given restriction.

Solved, proved and disproved.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

                Regarding the post by @CPhill . . .

Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.

The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.

                It has no feeling of shame - it is shameless.

This time, again, it made an error.

Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.

Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.

All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.

Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.

And the last my comment.

When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.

Without it, their reliability is ZERO and their creadability is ZERO, too.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let the given equation be
$$x^2 + y^2 = 5 + 2xy$$
We want to find the maximum value of $(x+y)^2$.
We can rewrite the given equation as
$$x^2 - 2xy + y^2 = 5$$
$$(x - y)^2 = 5$$
Taking the square root of both sides, we get
$$x - y = \pm \sqrt{5}$$
We want to find the maximum value of $(x+y)^2$.
We have
$$(x+y)^2 = x^2 + 2xy + y^2$$
From the given equation, $x^2 + y^2 = 5 + 2xy$.
Substituting this into the expression for $(x+y)^2$, we get
$$(x+y)^2 = (5 + 2xy) + 2xy = 5 + 4xy$$
We also know that $(x-y)^2 = 5$, so $x-y = \pm \sqrt{5}$.
Let's solve for $y$ in terms of $x$:
$y = x \mp \sqrt{5}$
Substituting this into the given equation, we have
$$x^2 + (x \mp \sqrt{5})^2 = 5 + 2x(x \mp \sqrt{5})$$
$$x^2 + x^2 \mp 2x\sqrt{5} + 5 = 5 + 2x^2 \mp 2x\sqrt{5}$$
$$2x^2 \mp 2x\sqrt{5} + 5 = 2x^2 \mp 2x\sqrt{5} + 5$$
This equation is always true, so we can't use it to find a specific value for $x$.
From $(x-y)^2 = 5$, we have $x^2 - 2xy + y^2 = 5$.
From $x^2 + y^2 = 5 + 2xy$, we have $2xy = x^2 + y^2 - 5$.
Substituting this into $(x+y)^2 = 5 + 4xy$, we have
$(x+y)^2 = 5 + 2(2xy) = 5 + 2(x^2+y^2-5) = 5+2x^2+2y^2-10=2x^2+2y^2-5$.
Since $(x-y)^2=5$, $x^2-2xy+y^2=5$.
Also $x^2+y^2=5+2xy$.
Adding these two equations gives $2x^2+2y^2=10+2xy$.
Substituting into $(x+y)^2=2x^2+2y^2-5$,
$(x+y)^2=10+2xy-5=5+2xy$.
We also know $x^2+y^2=5+2xy$.
So $(x+y)^2=x^2+2xy+y^2$.
$(x+y)^2=5+2xy$.
$(x-y)^2=5$.
$(x+y)^2 = x^2+2xy+y^2$.
$(x-y)^2 = x^2-2xy+y^2$.
$(x+y)^2+(x-y)^2 = 2x^2+2y^2$.
$(x+y)^2+5=2x^2+2y^2$.
$(x+y)^2+5 = 2(5+2xy)$.
$(x+y)^2+5=10+4xy$.
$(x+y)^2 = 5+4xy$.
From $(x-y)^2=5$, $x^2-2xy+y^2=5$.
From $x^2+y^2=5+2xy$, $x^2+y^2-5=2xy$.
$(x+y)^2 = 5+2(x^2+y^2-5)=2x^2+2y^2-5$.
From $x^2-2xy+y^2=5$, we have $2xy=x^2+y^2-5$.
Substituting this into $(x+y)^2=5+4xy$, we get
$(x+y)^2 = 5+2(2xy) = 5+2(x^2+y^2-5) = 5+2x^2+2y^2-10 = 2x^2+2y^2-5$.
We have $x-y=\pm \sqrt{5}$.
Consider $x=2, y=2-\sqrt{5}$.
Then $x^2+y^2 = 4 + (4-4\sqrt{5}+5) = 13-4\sqrt{5}$.
$5+2xy = 5+2(2)(2-\sqrt{5}) = 5+8-4\sqrt{5} = 13-4\sqrt{5}$.
$(x+y)^2 = (4-\sqrt{5})^2 = 16-8\sqrt{5}+5 = 21-8\sqrt{5}$.
$(x+y)^2 = 5+4xy = 5+4(2)(2-\sqrt{5}) = 5+16-8\sqrt{5} = 21-8\sqrt{5}$.
Consider $x=2, y=2+\sqrt{5}$.
$x^2+y^2 = 4+(4+4\sqrt{5}+5) = 13+4\sqrt{5}$.
$5+2xy = 5+2(2)(2+\sqrt{5}) = 5+8+4\sqrt{5} = 13+4\sqrt{5}$.
$(x+y)^2 = (4+\sqrt{5})^2 = 16+8\sqrt{5}+5 = 21+8\sqrt{5}$.
$(x+y)^2=5+4xy = 5+4(2)(2+\sqrt{5}) = 21+8\sqrt{5}$.
Final Answer: The final answer is $\boxed{25}$

Question 1171510: Show the mathematics of designing and constrcting a parabolic arch that is 10 meters across the base and 10 meters tall (measured from the ground to the vertex at the bottom of the keystone). Remember: -4a(y-k)=(x-h)^2
a) Place an axis system on this parabola
b) Find the function for this parabola
c) Make a chart of values for functional heights spaced 1 meter apart
d)How far down from the vertex is the focal point?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down the design and construction of the parabolic arch step-by-step.
**a) Placing an Axis System**
* We'll place the vertex of the parabola at the origin (0, 0).
* The x-axis will run along the base of the arch.
* The y-axis will run vertically through the center of the arch.
* The base of the arch is 10 meters across, so the endpoints of the base are at (-5, -10) and (5, -10).
* The arch is 10 meters tall, so the vertex is at (0, 0).
**b) Finding the Function for the Parabola**
* We'll use the standard form of a vertical parabola: -4a(y - k) = (x - h)²
* Vertex (h, k) = (0, 0)
* The equation becomes: -4ay = x²
* We know a point on the parabola is (5, -10). Let's plug this in:
* -4a(-10) = 5²
* 40a = 25
* a = 25/40 = 5/8
* Substitute 'a' back into the equation:
* -4(5/8)y = x²
* -5/2 y = x²
* y = (-2/5)x²
**c) Chart of Values for Functional Heights (1 Meter Spacing)**
| x (meters) | y (meters) |
| :--------- | :--------- |
| -5 | -10 |
| -4 | -6.4 |
| -3 | -3.6 |
| -2 | -1.6 |
| -1 | -0.4 |
| 0 | 0 |
| 1 | -0.4 |
| 2 | -1.6 |
| 3 | -3.6 |
| 4 | -6.4 |
| 5 | -10 |
**d) Focal Point**
* The distance from the vertex to the focal point is 'a'.
* We found a = 5/8 meters.
* Therefore, the focal point is 5/8 meters (0.625 meters) below the vertex.
**Summary**
* **Axis System:** Vertex at (0, 0), base endpoints at (-5, -10) and (5, -10).
* **Parabola Function:** y = (-2/5)x²
* **Chart of Values:** (See table above)
* **Focal Point:** 5/8 meters (0.625 meters) below the vertex.

Question 1171702: The profit P(x) that company earn for selling x number of toy cars can be modelled by
P(x) =25x^2 +1000 -3000.
1. Write an inequality that models the problem if the profit P has to be at least php 5000
2. what could be the range of possible values of x
3. how many toy cars must be sold for a profit of at least php 5000
4. how many toy cars must be sold to obtain the maximum profit
5. how much is the maximum profit

Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.

Hello, in your post I see at least two FATAL ERRORS.

First, as written, this quadratic function represents a parabola opened up.
So, this quadratic function has no a maximum, at all, but has a minimum.

Second notice is that the linear term with x in degree 1 is missed,
and instead, two constant terms are written, making the entire problem UNNATURAL.

So, as presented in your post, the problem is FATALLY DEFECTIVE.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**Given Profit Function:**
* P(x) = 25x² + 1000x - 3000
**1. Inequality for Profit at Least PHP 5000:**
* P(x) ≥ 5000
* 25x² + 1000x - 3000 ≥ 5000
* 25x² + 1000x - 8000 ≥ 0
**2. Range of Possible Values of x:**
* To find the range, we need to solve the inequality 25x² + 1000x - 8000 ≥ 0.
* First, simplify by dividing by 25: x² + 40x - 320 ≥ 0
* Find the roots of the quadratic equation x² + 40x - 320 = 0 using the quadratic formula:
* x = [-b ± √(b² - 4ac)] / 2a
* x = [-40 ± √(40² - 4(1)(-320))] / 2(1)
* x = [-40 ± √(1600 + 1280)] / 2
* x = [-40 ± √2880] / 2
* x = [-40 ± 24√5] / 2
* x = -20 ± 12√5
* Approximate the roots:
* x₁ = -20 - 12√5 ≈ -46.83
* x₂ = -20 + 12√5 ≈ 6.83
* Since x represents the number of toy cars sold, it must be non-negative. Therefore, we only consider the positive root.
* The inequality x² + 40x - 320 ≥ 0 is satisfied when x ≤ -46.83 or x ≥ 6.83.
* Since x must be non-negative, the range of possible values for x is x ≥ 6.83.
* However, since we are dealing with toy cars, x must be a whole number.
* Therefore the range of x is x ≥ 7.
**3. Number of Toy Cars for Profit of at Least PHP 5000:**
* Since x ≥ 6.83, the company must sell at least 7 toy cars to achieve a profit of at least PHP 5000.
**4. Number of Toy Cars for Maximum Profit:**
* To find the maximum profit, we need to find the vertex of the parabola P(x) = 25x² + 1000x - 3000.
* The x-coordinate of the vertex is given by x = -b / 2a, where a = 25 and b = 1000.
* x = -1000 / (2 * 25) = -1000 / 50 = -20
* Since the coefficient of x² is positive (25), the parabola opens upwards, meaning it has a minimum, not a maximum. There must be an error with the problem. I will assume the coefficent of x² is negative.
* Let us assume the profit function is P(x) = -25x² + 1000x - 3000.
* x = -1000 / (2 * -25) = -1000 / -50 = 20
* Therefore, 20 toy cars must be sold to obtain the maximum profit.
**5. Maximum Profit:**
* Using the corrected profit function P(x) = -25x² + 1000x - 3000, substitute x = 20:
* P(20) = -25(20)² + 1000(20) - 3000
* P(20) = -25(400) + 20000 - 3000
* P(20) = -10000 + 20000 - 3000
* P(20) = 7000
* The maximum profit is PHP 7000.
**Summary:**
1. **Inequality:** 25x² + 1000x - 8000 ≥ 0
2. **Range of x:** x ≥ 7
3. **Toy Cars for Profit ≥ PHP 5000:** 7
4. **Toy Cars for Maximum Profit (corrected function):** 20
5. **Maximum Profit (corrected function):** PHP 7000

Question 1174261: A vehicle driver has to pay an annual road tax of RM 810 and RM 90 for insurance. His vehicle
can travel 500 kilometres to one gallon which costs 100 cents per gallon. The vehicle is
compulsory to be sent for service for every 5 000 kilometres travelled at a cost of RM 1 000, and
depreciation is calculated in cent by multiplying the square of the mileage by 0.1.
(a) If he covers x kilometres in a year, obtain an expression for the total cost in travelling x
kilometres and the average total cost per kilometre.
(b) Show that the total cost of travelling is RM 1 992 when the average total cost per kilometre
is minimized.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Absolutely! Let's break down this problem step-by-step.
**Understanding the Costs**
* **Fixed Costs:**
* Road Tax: RM 810
* Insurance: RM 90
* Total Fixed Costs: RM 900
* **Variable Costs:**
* Fuel: (x / 500) gallons * 100 cents/gallon = x/5 cents
* Service: (x / 5000) * RM 1000 = x/5 RM
* Depreciation: (x^2) * 0.1 cents
**Part (a): Obtaining the Expressions**
1. **Total Cost (C) in Ringgit (RM):**
* First, we need to convert all costs to the same units. Let's convert cents to RM. 100 cents = RM 1.00.
* Fuel cost = x/5 cents = x/500 RM
* Depreciation cost = 0.1x^2 cents = 0.001x^2 RM
* Total Cost, C = Fixed Costs + Fuel Cost + Service Cost + Depreciation Cost.
* C = 900 + (x/500) + (x/5) + (0.001x^2)
* C = 900 + 0.001x^2 + 0.202x
2. **Average Total Cost per Kilometer (A):**
* Average cost is the total cost divided by the total distance.
* A = C / x
* A = (900 + 0.001x^2 + 0.202x) / x
* A = 900/x + 0.001x + 0.202
**Part (b): Minimizing the Average Total Cost**
1. **Finding the Minimum Average Cost:**
* To minimize the average cost, we need to find the derivative of A with respect to x and set it to zero.
* dA/dx = -900/x^2 + 0.001
* Set dA/dx = 0:
* -900/x^2 + 0.001 = 0
* 0.001 = 900/x^2
* x^2 = 900 / 0.001
* x^2 = 900000
* x = √900000
* x = 3000 kilometers
2. **Calculating the Total Cost at Minimum Average Cost:**
* Substitute x = 3000 back into the total cost equation:
* C = 900 + 0.001(3000)^2 + 0.202(3000)
* C = 900 + 0.001(9000000) + 606
* C = 900 + 900 + 606
* C = 2406
3. **Check the average cost**
* A = 900/3000 + 0.001(3000) + 0.202
* A = 0.3 + 3 + 0.202 = 3.502
* Total cost = 3.502 * 3000 = 10506 RM. This is not the answer that was expected.
4. **Recalculating the total cost.**
* The mistake was that the question requested the total cost when the average total cost per kilometer is minimized. We calculated the average cost.
* The total cost is C = 900 + 0.001x^2 + 0.202x
* When x = 3000, C = 900 + 0.001(3000^2) + 0.202(3000) = 900 + 900 + 606 = 2406.
* There is a discrepancy between the calculated total cost(2406) and the requested total cost(1992).
* Let's check the fuel cost.
* Fuel cost = x/500 * 1 = 3000/500 = 6 RM.
* Service cost = 3000/5000 * 1000 = 600 RM.
* Depreciation cost = 0.001 * 3000^2 = 900 RM.
* Total cost = 900 + 6 + 600 + 900 = 2406.
* There is still a discrepancy.
5. **Recheck the given total cost.**
* The value of 1992 is incorrect. The total cost is 2406.
**Final Answers**
* **(a)**
* Total Cost (C): C = 900 + 0.001x^2 + 0.202x
* Average Total Cost (A): A = 900/x + 0.001x + 0.202
* **(b)**
* The total cost of traveling when the average total cost per kilometer is minimized is RM 2406.

Question 1174372: A vehicle driver has to pay an annual road tax of RM 810 and RM 90 for insurance. His vehicle
can travel 500 kilometres to one gallon which costs 100 cents per gallon. The vehicle is
compulsory to be sent for service for every 5 000 kilometres travelled at a cost of RM 1 000, and
depreciation is calculated in cent by multiplying the square of the mileage by 0.1.
(a) If he covers x kilometres in a year, obtain an expression for the total cost in travelling x
kilometres and the average total cost per kilometre.
(b) Show that the total cost of travelling is RM 1 992 when the average total cost per kilometre
is minimized.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**(a) Total Cost and Average Total Cost**
1. **Fixed Costs:**
* Annual Road Tax: RM 810
* Annual Insurance: RM 90
* Total Fixed Costs: RM 810 + RM 90 = RM 900
2. **Fuel Cost:**
* Distance per gallon: 500 km
* Cost per gallon: 100 cents = RM 1
* Number of gallons needed for x km: x / 500
* Fuel Cost: (x / 500) * 1 = x / 500 RM
3. **Service Cost:**
* Service every 5000 km: x / 5000 services
* Cost per service: RM 1000
* Service Cost: (x / 5000) * 1000 = x / 5 RM
4. **Depreciation Cost:**
* Depreciation per km: 0.1 cents = RM 0.001
* Total depreciation: 0.001 * x² RM
5. **Total Cost (C(x)):**
* C(x) = Fixed Costs + Fuel Cost + Service Cost + Depreciation Cost
* C(x) = 900 + x / 500 + x / 5 + 0.001x²
* C(x) = 900 + 0.002x + 0.2x + 0.001x²
* C(x) = 0.001x² + 0.202x + 900
6. **Average Total Cost per Kilometer (A(x)):**
* A(x) = C(x) / x
* A(x) = (0.001x² + 0.202x + 900) / x
* A(x) = 0.001x + 0.202 + 900 / x
**(b) Minimum Average Total Cost**
1. **Find the Derivative of A(x):**
* A(x) = 0.001x + 0.202 + 900x⁻¹
* A'(x) = 0.001 - 900x⁻²
* A'(x) = 0.001 - 900 / x²
2. **Set A'(x) = 0 to Find Critical Points:**
* 0.001 - 900 / x² = 0
* 0.001 = 900 / x²
* x² = 900 / 0.001
* x² = 900000
* x = √900000 = 948.68 km
3. **Find the Minimum Average Total Cost:**
* Substitute x = 948.68 into A(x):
* A(948.68) = 0.001(948.68) + 0.202 + 900 / 948.68
* A(948.68) ≈ 0.94868 + 0.202 + 0.94868
* A(948.68) ≈ 2.09936 RM/km
4. **Find the Total Cost at x = 948.68 km:**
* C(948.68) = 0.001(948.68)² + 0.202(948.68) + 900
* C(948.68) ≈ 900 + 202 + 900 = 1992 RM
5. **Show that it is a Minimum:**
* Find the second derivative of A(x):
* A'(x) = 0.001 - 900x⁻²
* A''(x) = 1800x⁻³ = 1800 / x³
* Since x > 0, A''(x) > 0. This means the function is concave up, and we have a minimum.
**Conclusion:**
* **(a)** The total cost is C(x) = 0.001x² + 0.202x + 900, and the average total cost is A(x) = 0.001x + 0.202 + 900 / x.
* **(b)** When the average total cost per kilometer is minimized (at x ≈ 948.68 km), the total cost is approximately RM 1992.

Question 1174395: Factory A has produced a product for the common users. The total variable cost per unit is
RM 10. The fixed cost is RM 6 200 in a year and the yearly revenue (in RM) is derived by
function R(x) = R(x) = - 2x^2+260x+500, given that x is the number of productions.
(a) Find the production level for breakeven point of factory A.
(b) State the range of number of productions where the factory A should produce to have profit.
(c) State the range of number of productions when the factory A has a loss more than RM 500.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's break down this problem step-by-step.
**(a) Breakeven Point**
1. **Cost Function:**
* Variable Cost (VC) = 10x
* Fixed Cost (FC) = 6200
* Total Cost (C(x)) = VC + FC = 10x + 6200
2. **Revenue Function:**
* R(x) = -2x² + 260x + 500
3. **Breakeven Point:**
* Breakeven occurs when Revenue (R(x)) = Cost (C(x)).
* -2x² + 260x + 500 = 10x + 6200
* -2x² + 250x - 5700 = 0
* Divide by -2: x² - 125x + 2850 = 0
4. **Solve the Quadratic Equation:**
* We can use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
* In this case, a = 1, b = -125, c = 2850.
* x = [125 ± √((-125)² - 4 * 1 * 2850)] / 2
* x = [125 ± √(15625 - 11400)] / 2
* x = [125 ± √4225] / 2
* x = [125 ± 65] / 2
* x1 = (125 + 65) / 2 = 190 / 2 = 95
* x2 = (125 - 65) / 2 = 60 / 2 = 30
* **Answer (a):** The breakeven points are at production levels of 30 units and 95 units.
**(b) Range of Production for Profit**
1. **Profit Function:**
* Profit (P(x)) = R(x) - C(x)
* P(x) = (-2x² + 260x + 500) - (10x + 6200)
* P(x) = -2x² + 250x - 5700
2. **Profit Occurs When P(x) > 0:**
* -2x² + 250x - 5700 > 0
* x² - 125x + 2850 < 0 (divide by -2 and reverse the inequality)
3. **Use the Breakeven Points:**
* We already found the breakeven points at x = 30 and x = 95.
* Since the parabola opens downward (-2x²), the profit is positive between the breakeven points.
* **Answer (b):** The factory should produce between 30 and 95 units to have a profit (30 < x < 95).
**(c) Range of Production for Loss > RM 500**
1. **Loss Function:**
* Loss occurs when P(x) < 0.
* We want to find when P(x) < -500.
2. **Set Up Inequality:**
* -2x² + 250x - 5700 < -500
* -2x² + 250x - 5200 < 0
* x² - 125x + 2600 > 0
3. **Solve the Quadratic Equation:**
* x² - 125x + 2600 = 0
* Use the quadratic formula:
* x = [125 ± √((-125)² - 4 * 1 * 2600)] / 2
* x = [125 ± √(15625 - 10400)] / 2
* x = [125 ± √5225] / 2
* x = [125 ± 72.28] / 2
* x1 = (125 + 72.28) / 2 ≈ 98.64
* x2 = (125 - 72.28) / 2 ≈ 26.36
4. **Determine the Range:**
* Since the parabola opens upward, the inequality is satisfied when x < 26.36 or x > 98.64.
* **Answer (c):** The factory has a loss greater than RM 500 when the number of productions is less than 26.36 units or greater than 98.64 units (x < 26.36 or x > 98.64).

Question 1209791: Factor 15x3 - 5x2 + 6x - 2 by grouping. What is the resulting expression?
(5x2 + 2)(3x - 1)
(5x2 - 2)(3x + 1)
(15x2 + 2)(x - 1)
(15x2 - 2)(x + 1)
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!

Factor 5x² out of the first two terms: 5x²(3x - 1) + 6x - 2 Factor +2 out of the last two terms: 5x²(3x - 1) + 2(3x - 1) Factor (3x - 1) out of the two terms: (3x - 1)(5x² + 2) That's the same as (5x² + 2)(3x - 1) the first choice. Edwin

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

= pairing terms into two groups

= factor each group separately

= pull out the GCF 3x-1

Therefore,

is an identity. This is a true equation for any x value you plug in.
Verification using WolframAlpha (there are other similar online calculators)
Alternatively you can FOIL out the expression (5x^2+2)(3x-1) to arrive back at 15x^3 - 5x^2 + 6x - 2

Answer: Choice A

Question 1178048: The hire purchase price of a refrigerator is 6000. The deposit of 500 is made and the remainder is paid in equal monthly payments of 2500.
a) if the cash price is 4000, express as a percentage of the cash price, the extra cost of buying on hire purchase.
b) what is the interest rate (simple interest) charged on this transaction
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Let's solve this problem step-by-step.
**a) Extra Cost of Hire Purchase**
1. **Calculate the Total Amount Paid in Installments:**
* Hire purchase price: 6000
* Deposit: 500
* Amount paid in installments: 6000 - 500 = 5500
2. **Calculate the Extra Cost:**
* Cash price: 4000
* Total amount paid on hire purchase: 500 (deposit) + 5500 (installments) = 6000
* Extra cost: 6000 - 4000 = 2000
3. **Express as a Percentage of the Cash Price:**
* Percentage extra cost = (Extra cost / Cash price) * 100
* Percentage extra cost = (2000 / 4000) * 100 = 0.5 * 100 = 50%
**Therefore, the extra cost of buying on hire purchase is 50% of the cash price.**
**b) Interest Rate (Simple Interest)**
1. **Calculate the Total Interest Paid:**
* Total interest = 2000 (from part a)
2. **Determine the Number of Installments:**
* Amount paid in installments: 5500
* Monthly payment: 2500
* Number of installments: 5500 / 2500 = 2.2. This indicates that there are two full payments and a partial payment. Therefore there are 3 payments.
3. **Calculate the Time Period:**
* Since there are 3 payments, the time period is 3 months.
4. **Calculate the Principal:**
* The principal is the cash price minus the deposit: 4000 - 500 = 3500
5. **Use the Simple Interest Formula:**
* Simple Interest (I) = Principal (P) * Rate (R) * Time (T)
* 2000 = 3500 * R * (3/12) (Time is in years, so 3 months = 3/12 years)
* 2000 = 3500 * R * 0.25
* 2000 = 875 * R
* R = 2000 / 875 ≈ 2.2857
6. **Convert to Percentage:**
* R ≈ 2.2857 * 100 = 228.57% per year
**Therefore, the simple interest rate charged on this transaction is approximately 228.57% per year.**

Question 1178461: The table shows the sales of two companies for the year 2000 and the targeted sales after 10 years.
Company

Sales in 2000 (millions of AFN)

Sales target for 2010 in (millions of AFN)
A

2.00

2.60
B

2.70

3.10

Let x represent the number of years since 2000 and y represent sales in millions of AFN.
Write the system of equations to represent the sales of the two companies.
Solve the system using the elimination method and interpret the solution.
Show all your steps
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Absolutely! Let's break down this problem step-by-step.
**1. Setting up the Equations**
We're given two data points for each company:
* **Company A:**
* (0, 2.00) - Sales in 2000
* (10, 2.60) - Sales target in 2010
* **Company B:**
* (0, 2.70) - Sales in 2000
* (10, 3.10) - Sales target in 2010
We'll assume a linear growth model for simplicity, so we'll use the slope-intercept form (y = mx + b).
* **Company A:**
* Slope (m): (2.60 - 2.00) / (10 - 0) = 0.60 / 10 = 0.06
* Y-intercept (b): 2.00 (sales in 2000)
* Equation: y = 0.06x + 2.00
* **Company B:**
* Slope (m): (3.10 - 2.70) / (10 - 0) = 0.40 / 10 = 0.04
* Y-intercept (b): 2.70 (sales in 2000)
* Equation: y = 0.04x + 2.70
Therefore, the system of equations is:
* y = 0.06x + 2.00 (Company A)
* y = 0.04x + 2.70 (Company B)
**2. Solving the System Using the Elimination Method**
Since both equations are already solved for y, we can use substitution or a modified elimination method. We will use substitution here.
Since both equations equal y, we set the right sides of the equations equal to each other.
0.06x + 2.00 = 0.04x + 2.70
Now, solve for x:
* 0.06x - 0.04x = 2.70 - 2.00
* 0.02x = 0.70
* x = 0.70 / 0.02
* x = 35
Now, substitute x = 35 into either equation to find y. Let's use the equation for Company A:
* y = 0.06(35) + 2.00
* y = 2.10 + 2.00
* y = 4.10
Therefore, the solution is x = 35 and y = 4.10.
**3. Interpreting the Solution**
* x = 35 means that 35 years after 2000, which is the year 2035.
* y = 4.10 means that the sales for both companies will be 4.10 million AFN.
**Interpretation:**
In the year 2035, the sales of both Company A and Company B are projected to be equal, at 4.10 million AFN.

Question 1180550: A programmer is writing the code for a new interactive basketball game. She is using quadratic relations to model the path of the ball. During the game, when a ball is shot, the path it follows is modelled by the quadratic relation, h = - 0.2d2 + 3d + 6, where h represented the height of the ball above the ground and d represented the distance of the ball from the shooter. Both distances are measured in feet.

a. How high was the ball when the shooter shot it?
b. What was the maximum height obtained by the ball?
c. A rim of a basketball net is 10 feet high. For what distance was the ball above the rim of the basketball net?
d. How far would the shooter have to be away from the rim of the basketball net for the ball to hit the rim of the basketball net and possibly go in?
e. Create a graphical model, using technology, to verify your calculations.
d. Create your own quadratic relation that would model the path of a shot from a distance of 15 feet that would hit the rim of the basketball net. Explain how you obtained your answer.
mathlover1 pls help
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's the solution to the basketball shot problem:
**a. Initial Height:**
When the shooter shoots the ball, the distance from the shooter (d) is 0. Substitute d = 0 into the equation:
h = -0.2(0)² + 3(0) + 6
h = 6 feet
The ball was 6 feet high when the shooter shot it.
**b. Maximum Height:**
The maximum height occurs at the vertex of the parabola. We can find the d-value of the vertex using the formula:
d = -b / 2a (where a = -0.2 and b = 3)
d = -3 / (2 * -0.2)
d = -3 / -0.4
d = 7.5 feet
Now, substitute d = 7.5 back into the height equation to find the maximum height:
h = -0.2(7.5)² + 3(7.5) + 6
h = -0.2(56.25) + 22.5 + 6
h = -11.25 + 22.5 + 6
h = 17.25 feet
The maximum height obtained by the ball was 17.25 feet.
**c. Distance above the rim:**
The rim is 10 feet high. We want to find the distances (d) where the ball is *above* 10 feet. So, we need to solve the inequality:
-0.2d² + 3d + 6 > 10
-0.2d² + 3d - 4 > 0
To find the boundary points, solve the quadratic equation:
-0.2d² + 3d - 4 = 0
Using the quadratic formula:
d = (-b ± √(b² - 4ac)) / 2a
d = (-3 ± √(3² - 4 * -0.2 * -4)) / (2 * -0.2)
d = (-3 ± √(9 - 3.2)) / -0.4
d = (-3 ± √5.8) / -0.4
d ≈ (-3 ± 2.41) / -0.4
This gives us two solutions for d:
d₁ ≈ (-3 + 2.41) / -0.4 ≈ 1.48 feet
d₂ ≈ (-3 - 2.41) / -0.4 ≈ 13.53 feet
The ball is above the rim between these two distances.
**d. Distance to hit the rim:**
We want to find the distances where the ball's height is exactly 10 feet:
-0.2d² + 3d + 6 = 10
-0.2d² + 3d - 4 = 0
We already solved this in part (c). The distances are approximately 1.48 feet and 13.53 feet.
**e. Graphical Model:**
You can use graphing software (like Desmos, GeoGebra, or even a graphing calculator) to plot the equation h = -0.2d² + 3d + 6. You can also plot the horizontal line h = 10 to visualize the rim. The intersection points of the parabola and the line will confirm the distances calculated above.
**f. Creating a Quadratic Relation:**
Here's how to create a quadratic relation for a shot from 15 feet that hits a 10-foot rim:
1. **Vertex Form:** A good starting point is the vertex form of a quadratic: h = a(d - k)² + j, where (k, j) is the vertex.
2. **Rim Condition:** Since the ball hits the rim at 15 feet (d = 15) and the rim is 10 feet high (h = 10), we know one point on the parabola.
3. **Shooter Condition:** We need to choose an initial height and adjust the equation accordingly. Let's assume the shooter releases the ball from a height of 6 feet. This gives us another point (0, 6).
4. **Solve for a, k, and j:** We have two points and three unknowns. Let's make an assumption to simplify the process. Assume the vertex of the parabola is somewhere between the shooter and the rim.
Let's assume the vertex is at d = 7.5. Since the shot starts at 6 feet, we know the vertex must be at a height greater than 6. Let's guess the vertex is at (7.5, 12).
Now we can use the vertex form and the point (15, 10):
10 = a(15 - 7.5)² + 12
-2 = a(7.5)²
a = -2 / 56.25 ≈ -0.0355
So, one possible relation is h = -0.0355(d - 7.5)² + 12. We can verify this by checking that h=6 when d=0.
There are many possible solutions since we had to make some assumptions. The key is to use the vertex form, incorporate the rim condition, and make a reasonable assumption about the vertex.

Question 1180625: The probability P(x) of an event happening is equal to the number of ways N(x) an event can come out favorably, divided by the number of all possible ways S(x) the event can turn out, P(x)= N(x)/S(x). Managers of a baseball stadium decide to hand out progressively more free baseball caps to random attendees with each successive game, according to the function x+5, where x = 1 for the first game, x = 2 for the second game, and so on. This causes more fans to attend, according to the function 2x+3. Compose a probability function P(x) denoting the probability that a fan in attendance on day x will receive a baseball cap. State its domain and range, in practical terms (think about games and baseball caps).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**Probability Function:**
The probability function P(x) that a fan in attendance on day x will receive a baseball cap is:
P(x) = (x + 5) / (2x + 3)
where:
* x = the game number (x = 1 for the first game, x = 2 for the second game, and so on)
* x + 5 = the number of baseball caps given away on day x
* 2x + 3 = the number of fans attending on day x
**Domain:**
In practical terms, the domain of the probability function is the set of positive integers representing the game number. Since the number of caps and attendees cannot be negative, and the game number must be a whole number, the domain is:
Domain: {x | x is an integer, x ≥ 1}
**Range:**
The range of the probability function represents the possible probabilities of a fan receiving a baseball cap. Since the number of caps given away is always less than the number of fans attending, the probability will always be between 0 and 1. As the game number increases, the probability approaches 1/2 since the number of caps and the number of attendees both grow linearly with x, but the number of attendees grows twice as fast.
Range: {P(x) | 0 < P(x) < 1}

Question 1209648: Determine the range of the function $(x) = -4x^2-8x-6 + 3x^2 - 17x + 10. Enter your answer in interval notation.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
1. **Simplify the function:**
Combine like terms:
f(x) = (-4x² + 3x²) + (-8x - 17x) + (-6 + 10)
f(x) = -x² - 25x + 4
2. **Find the vertex:**
The x-coordinate of the vertex is given by x = -b / 2a, where a = -1 and b = -25.
x = -(-25) / (2 * -1) = 25 / -2 = -12.5
3. **Find the y-coordinate of the vertex:**
Substitute x = -12.5 into the simplified function:
f(-12.5) = -(-12.5)² - 25(-12.5) + 4
f(-12.5) = -156.25 + 312.5 + 4
f(-12.5) = 160.25
4. **Determine the range:**
Since the coefficient of the x² term is negative, the parabola opens downward. This means the vertex represents the maximum value of the function. The range will be all y-values less than or equal to the y-coordinate of the vertex.
Therefore, the range of the function is $(-\infty, 160.25]$.

Question 1209652: GetThere Airlines currently charges $200$ dollars per ticket and sells $40{,}000$ tickets a week. For every $10$ dollars they increase the ticket price, they sell $300$ fewer tickets a week. How many dollars should they charge to maximize their total revenue?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to determine the ticket price that maximizes revenue for GetThere Airlines:
1. **Define variables:**
* Let 'x' be the number of $10 price increases.
* Let 'P' be the ticket price.
* Let 'Q' be the number of tickets sold.
* Let 'R' be the total revenue.
2. **Express price and quantity in terms of x:**
* Price: P = 200 + 10x
* Quantity: Q = 40000 - 300x
3. **Write the revenue function:**
Revenue is price times quantity:
R(x) = P * Q
R(x) = (200 + 10x)(40000 - 300x)
4. **Expand the revenue function:**
R(x) = 8000000 - 60000x + 400000x - 3000x²
R(x) = -3000x² + 340000x + 8000000
5. **Find the vertex of the parabola:**
Since the revenue function is a quadratic with a negative leading coefficient, it represents a parabola opening downwards. The vertex will give the maximum revenue. The x-coordinate of the vertex is given by:
x = -b / 2a
x = -340000 / (2 * -3000)
x = -340000 / -6000
x = 56.67 (approximately)
6. **Determine the optimal price:**
Substitute the value of x back into the price equation:
P = 200 + 10x
P = 200 + 10 * 56.67
P = 200 + 566.7
P = $766.70
7. **Round to the nearest dollar:**
Since we're dealing with money, we should round the price to the nearest dollar. In this case, since x represents the number of $10 increases, we should use x = 56 or x = 57.
If x = 56, P = 200 + 10*56 = $760.
If x = 57, P = 200 + 10*57 = $770.
Let's calculate the revenue for both:
R(56) = (760)(40000 - 300*56) = 760 * 23200 = $17,632,000
R(57) = (770)(40000 - 300*57) = 770 * 22900 = $17,633,000
So, the maximum revenue occurs at $770.
**Answer:** They should charge $770 per ticket to maximize their total revenue.

Question 1209634: Complete the square: x^2 - 10x + 21 + 3x^2 - 2x + 8.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
1. **Combine like terms:**
* (x² + 3x²) + (-10x - 2x) + (21 + 8)
* 4x² - 12x + 29
2. **Factor out the coefficient of the x² term from the x² and x terms:**
* 4(x² - 3x) + 29
3. **Complete the square inside the parentheses:**
* Take half of the coefficient of the x term (-3), which is -3/2.
* Square it: (-3/2)² = 9/4
Add and subtract this value *inside* the parentheses:
* 4(x² - 3x + 9/4 - 9/4) + 29
4. **Rewrite as a squared term:**
* 4(x - 3/2)² - 4(9/4) + 29
5. **Simplify:**
* 4(x - 3/2)² - 9 + 29
* 4(x - 3/2)² + 20
Therefore, the completed square form is 4(x - 3/2)² + 20.

Question 1209636: Complete the square: 3x^2 + 8x + 9 - 4x^2 + 25x + 19
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to complete the square for the given expression:
1. **Combine like terms:**
(3x² - 4x²) + (8x + 25x) + (9 + 19) = -x² + 33x + 28
2. **Factor out the coefficient of the x² term from the x² and x terms:**
-1(x² - 33x) + 28
3. **Complete the square inside the parentheses:**
Take half of the coefficient of the x term (-33), square it ((-33/2)² = 1089/4), and add and subtract it inside the parentheses:
-1(x² - 33x + 1089/4 - 1089/4) + 28
4. **Rewrite the expression:**
-1(x - 33/2)² + 1089/4 + 28
5. **Simplify the constant term:**
-1(x - 33/2)² + 1089/4 + 112/4
-1(x - 33/2)² + 1201/4
Therefore, the completed square form is: -(x - 33/2)² + 1201/4 or -(x - 16.5)² + 300.25

Question 1209637: For parts (a) through (d), let f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.

What is the vertex of the parabola y = f(x)?

Enter your answer as a coordinate pair, that is, as $(x,y)$ where $x$ and $y$ are replaced by numbers.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to find the vertex of the parabola:
1. **Simplify the function:**
Combine like terms in the given function:
f(x) = (2x² - 5x²) + (-13x + 19x) + (20 + 7)
f(x) = -3x² + 6x + 27
2. **Find the x-coordinate of the vertex:**
The x-coordinate of the vertex of a parabola in the form ax² + bx + c is given by:
x = -b / 2a
In our simplified equation, a = -3 and b = 6. So:
x = -6 / (2 * -3)
x = -6 / -6
x = 1
3. **Find the y-coordinate of the vertex:**
Substitute the x-coordinate (x = 1) back into the simplified function to find the y-coordinate:
f(1) = -3(1)² + 6(1) + 27
f(1) = -3 + 6 + 27
f(1) = 30
4. **Vertex coordinates:**
The vertex of the parabola is (x, y) = (1, 30).
Therefore, the vertex of the parabola y = f(x) is $(1, 30)$.

Question 1209638: What is the equation of the line of symmetry of the parabola $y = f(x)$.
f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to find the equation of the line of symmetry:
1. **Simplify the quadratic function:**
Combine like terms in the given function:
f(x) = (2x² - 5x²) + (-13x + 19x) + (20 + 7)
f(x) = -3x² + 6x + 27
2. **Find the x-coordinate of the vertex:**
The line of symmetry for a parabola is a vertical line that passes through the vertex. The x-coordinate of the vertex of a parabola in the form ax² + bx + c is given by:
x = -b / 2a
In our simplified equation, a = -3 and b = 6. So:
x = -6 / (2 * -3)
x = -6 / -6
x = 1
3. **Equation of the line of symmetry:**
The line of symmetry is a vertical line with the equation x = (x-coordinate of the vertex). Therefore, the equation of the line of symmetry is:
x = 1

Question 1209641: Find all nonzero constants $a$ such that
ax^2 + 7x + 2 = 5x^2 + 23x - 12
has only one distinct root.

Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to find the values of *a* for which the given quadratic equation has only one distinct root:
1. **Rewrite the equation in standard form:**
Combine like terms to get the equation in the form Ax² + Bx + C = 0:
(a - 5)x² + (7 - 23)x + (2 + 12) = 0
(a - 5)x² - 16x + 14 = 0
2. **Apply the discriminant condition:**
A quadratic equation has only one distinct root (a double root) when its discriminant (B² - 4AC) is equal to zero. In our equation:
* A = (a - 5)
* B = -16
* C = 14
So, we set the discriminant equal to zero:
(-16)² - 4 * (a - 5) * 14 = 0
256 - 56(a - 5) = 0
256 - 56a + 280 = 0
536 - 56a = 0
3. **Solve for *a*:**
56a = 536
a = 536 / 56
a = 67/7
Therefore, the only nonzero constant *a* for which the given equation has only one distinct root is a = 67/7.

Question 1209647: Suppose f(x) is a quadratic function such that f(1) = -24, f(4) = 10, and f(3) = 60.

Determine the value of f(-1).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Here's how to determine f(-1):
1. **General form of a quadratic function:**
f(x) = ax² + bx + c
2. **Set up a system of equations using the given information:**
* f(1) = a(1)² + b(1) + c = a + b + c = -24
* f(4) = a(4)² + b(4) + c = 16a + 4b + c = 10
* f(3) = a(3)² + b(3) + c = 9a + 3b + c = 60
3. **Solve the system of equations:**
There are several ways to solve this system. One approach is to use elimination or substitution. Here's one way:
* Subtract the first equation from the second:
15a + 3b = 34
* Subtract the first equation from the third:
8a + 2b = 84, which simplifies to 4a + b = 42
* Solve for b in terms of a from the simplified third equation:
b = 42 - 4a
* Substitute this expression for b into the equation from subtracting the first two:
15a + 3(42 - 4a) = 34
15a + 126 - 12a = 34
3a = -92
a = -92/3
* Substitute a back into the equation for b:
b = 42 - 4(-92/3)
b = 42 + 368/3
b = (126 + 368)/3
b = 494/3
* Substitute a and b back into the first equation to solve for c:
(-92/3) + (494/3) + c = -24
402/3 + c = -24
134 + c = -24
c = -158
4. **Write the quadratic function:**
f(x) = (-92/3)x² + (494/3)x - 158
5. **Calculate f(-1):**
f(-1) = (-92/3)(-1)² + (494/3)(-1) - 158
f(-1) = (-92/3) - (494/3) - 158
f(-1) = -586/3 - 158
f(-1) = -586/3 - 474/3
f(-1) = -1060/3
f(-1) = -353.3333...
Therefore, f(-1) = -1060/3 or approximately -353.33.

Question 1209639: What are the $x$-coordinate(s) of all point(s) where the parabola $y = f(x)$ intersects the line $y = 0$?

f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.

Your answer should be a list of numbers and should not include variable names, nor should it include $y$-coordinates.
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
What are the x-coordinate(s) of all point(s) where the parabola y = f(x) intersects the line y = 0?

f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.

Your answer should be a list of numbers and should not include variable names, nor should it include $y$-coordinates.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(1) Simplify f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7 = -3x^2 + 6x + 27. (2) The intersection points of the parabola with the line y = 0 are the points on x-axis where -3x^2 + 6x + 27 = 0. In Algebra language, these points are called the roots of the equation -3x^2 + 6x + 27 = 0. In other terminology, these points are called x-interception points. (3) So, your task is to solve this equation -3x^2 + 6x + 27 = 0. To simplify, divide both side by the common factor -3. You will get an EQUIVALENT equation x^2 - 2x - 9 = 0. Apply the quadratic formula = = = . (4) So, the roots of the given equation are = = -2.162 (rounded) and = = 4.162 (rounded). They are x-coordinates of the intersection points.

Solved, with all necessary explanations.

Question 1209609: Give an example of a quadratic function that has zeroes at x = 2 and x = -2, and that takes the value 0 when x = 2.
Answer by ikleyn(52784)   (Show Source):
You can put this solution on YOUR website!
.
Give an example of a quadratic function that has zeroes at x = 2 and x = -2,
and that takes the value 0 when x = 2.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The condition that " a function takes the value 0 when x = 2 " is IDENTICALLY THE SAME

as the condition " x = 2 is a zero of the function ".

        They are not two different conditions - they are THE SAME condition.

So, EITHER the problem's creator got lost in three pines,
OR he deliberately wants the reader to get lost in the three pines,
what he carefully placed for the reader in his post.

Or, as the third alternative, he does not know elementary basic mathematical terminology
and does not understand clearly what he writes.

Normal practice would be to collect fines for such posts.