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Question 994712: If 321n=232seven.find n quadratically
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What is 232seven ?

Question 994539: Say whether the graph of the equation y=-x^2+4x-1 opens up, down, left, or right.Give the coordinates of the vertex and the equation of the axis of symmetry.Write your quadratic function in vertex form. Find the maximum or minimum value. Explain how you can determine this value.

Found 2 solutions by MathTherapy, josgarithmetic:
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Say whether the graph of the equation y=-x^2+4x-1 opens up, down, left, or right.Give the coordinates of the vertex and the equation of the axis of symmetry.Write your quadratic function in vertex form. Find the maximum or minimum value. Explain how you can determine this value.

----- Completed the square to obtain equation in vertex form
Opens concave down since the coefficient on  is < 0
Coordinates of the vertex: ()
Equation of the axis of symmetry:
Maximum value, or y-coordinate of vertex: , or



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Coefficient on leading term is . Parabola is concave down. (Opens downward).

Complete the Square to help change to standard form.

Vertex is (-2,5).

Question 994300: The perimeter of a backyard is 560 feet. If the width is 125 feet less than the length, find the length of the yard.
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The perimeter is: 2L+2W= 560 (1st formula)
And the problem says that:
W= L-125 Now apply the first formula using this value for W:
2L+2(L-125)= 560 Multiply on left
2L+2L-250= 560 Add L on left and add 250 to both sides:
4L= 810 Divide both sides by 4
L= 202.5 OK, we have our Length. And the problem says that the Width is 202.5-125= 77.5. Let's prove that we have the correct answer:
2L+2W= 560
2(202.5)+2(77.5)= 560
405+155= 560
560= 560 We have the right answer.

Question 994206: Solve the equation:
Ally inherits $65,000 and decides to invest part of it in an education account for her daughter and the rest in a 10-year CD. If the amount she puts in the education account is$12,000 more than twice the amount she puts in the CD, how much money does Ally invest in each account?

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Ally inherits $65,000 and decides to invest part of it in an education account for her daughter and the rest in a 10-year CD. If the amount she puts in the education account is$12,000 more than twice the amount she puts in the CD, how much money does Ally invest in each account?
-----------
In Ed. account: E
In CD account:: C
-----
Equations:
E + C = 65000
E - C = 12000
---
2E = 77000
E = 38500
C = 38500+12000 = 50500
-----------
Cheers,
Stan H.
-------------

Question 994153: The length of a rectangle is 10 cm more than its width. The area of the rectangle is 56 cm2.
Calculate the width of the rectangle.

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The length of a rectangle is 10 cm more than its width. The area of the rectangle is 56 cm2.
Calculate the width of the rectangle.
=============
Find a pair of factors of 56 that differ by 10.

Question 994128: (a)If the quadratic equation of 6x^2－7x＝2k has two distinct real roots,
(i) find the range of possible values of k,
(ii) find the roots of the equation when k is the smallest integer in (a)(i)
(b) Using the result of (a)(ii), solve －3(x＋p)^2＋7/2(x＋p)－5/6＝0, where p is a constant. (give the answers in terms of p.)

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For two distinct roots, the discriminant must be greater than zero, so:

John

My calculator said it, I believe it, that settles it

Question 994123: What is the answer if you derive a Q.F from table or graph given point (0,5),(4,1),(-3,13)
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John

My calculator said it, I believe it, that settles it

Question 994062: The edges of three cubes are consecutive odd integers. If the cubes are stacked on a deck, as shown, the total exposed surface area is 381. Find the lengths of the sides of the cubes.
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Shown where?

Question 994033: The graph of Quadratic equation of opens up if(?) and down if(?). Explain why or how that statement is true.
(Hint: consider what happens as x gets more and more positive or more and more negative.)

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f(x)=-x^2. This opens up downward. As x^2 gets larger, -x^2 gets larger negative, so the graph goes down with increasing positive or negative x.
f(x)=x^2, here, when x is positive or negative x^2 is always positive, like the first case. But there is no negative sign in front of the x^2, so the graph opens upward.

Question 994034: In general, why might it be important to solve an equation for different variables?
For example, the equation for degrees Fahrenheit is given by, where C is degrees Celsius. In this example, why—or when—would we want to solve for C? Feel free to discuss other examples.

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If you are in any other country in the world and say what the temperature is in Fahrenheit, they will not be able to understand what you mean. They use Celsius, and you have to convert it to Celsius.
If you are asked to give a medication in a dose of 18 mg/kg, and the person weighs 154 pounds, how many kg are in 154 pounds?
If you are in Europe, they measure fuel economy by number of liters of fuel it takes to go 100 km. In order to do this, you need to convert gallons to liters and miles to kilometers. These are a few of the examples, and this is just metric-English. There are many other equations in multiple variables that one may have to solve to determine, for example, what percentage of a change is due to normal variability and what is due to a change in the system itself.

Question 993886: Hi
I am given this equation, h= height (meters), t= time (seconds)
h=2t(50-t)
How long is the rocket in air?
-I just applied the null factor law for that equation, and got:
2t = 0
t=0
and
50-t=0
-t=-50
t=50
Does that mean the rocket was in the air for 50 seconds?
or did i do something wrong or did not take in to account
thank you

Found 2 solutions by Theo, Alan3354:
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it appears the rocket was in the air for 50 seconds.

your equation was h = 2t * (50 - t)

when you set it equal to 0, you find the x-intercepts.

those occur at t = 0 and t = 50.

since the coefficient of the leading term is negative, the quadratic equation points up and opens down.

to see this easier, simplify the equation by multiplying the factors out.

2t * (50 - t) = 100t - t^2

reorder the terms to get h = -t^2 + 100t

the leading coefficient is negative which makes the parabola (graph of a quadratic equation is a parabola) point up (maximum point is on top) and open down.

since all value of t have the graph positive between 0 and 50, that's the time that the graph of the equation is above the x-axis which represents the time the rocket is in the air.

here's what the graph of your equation looks like:

i used x instead of t because that's what the graphing software requires.

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I am given this equation, h= height (meters), t= time (seconds)
h=2t(50-t)
How long is the rocket in air?
-I just applied the null factor law for that equation, and got:
2t = 0
t=0
and
50-t=0
-t=-50
t=50
Does that mean the rocket was in the air for 50 seconds?
or did i do something wrong or did not take in to account
===========================
No, you got it right.
It's launched at t = 0 and impacts at t = 50.
=========================
PS This is not a rocket. Rockets have thrust engines and accelerate upward. This is just a projectile.

Question 993880: Hi
i am wondering if this quadratic expression can be factored in anyway:
3x^2-26
I tried

Not sure if this works :s

Found 2 solutions by Theo, Alan3354:
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yes, it will work.

the two equations are equivalent.

is equal to:

which is equal to:

if you graph both equations, they will be identical.

here's the graph of both equations on the same graph.

since the equations are identical, it looks like one graph.

this is the graphing statement enclosed in 3 opening brackets at the beginning and 3 closing brackets at the end.

i used [ instead of { and i used ] instead of } so you can see the formula easily without having to go to the source.

[[[graph(600,600,-10,10,-30,30,3*(x^2-26/3),3*(x-sqrt(26/3))*(x+sqrt(26/3)))]]]

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3x^2-26
I tried

Not sure if this works
--------------
What you did is not incorrect, but factoring implies using integers.

Question 993876: A 1.8 meters tall person walks across the path of the javelin. He is 58 meters away from where the javelin was launched. Will he be hit by the javelin? Justify your answer.
The quadratic equation i'm given is:
Do I replace 'x' with 58 and solve for y? so i know the height of the javelin at x=58 or 58 meters away to see if the javelin hit the person?
Like,

y ≈ 3.29m
Hence the javelin does not hit the person that is 1.8 meters tall.
Is this how i 'justify'?

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A 1.8 meters tall person walks across the path of the javelin. He is 58 meters away from where the javelin was launched. Will he be hit by the javelin? Justify your answer.
The quadratic equation i'm given is:
Do I replace 'x' with 58 and solve for y? so i know the height of the javelin at x=58 or 58 meters away to see if the javelin hit the person?
Like,

y ≈ 3.29m
Hence the javelin does not hit the person that is 1.8 meters tall.
Is this how i 'justify'?
-----------------
y gives the height of the javelin for the given x, so that justifies your answer.
should be

y =~3.16 meters
but that doesn't change the Yes or No answer.

Question 993728: 2x^2+3b-4=0
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.

Question 993631: Where do lines y = x - 1 and y = -5x + 11 intersect?
Found 2 solutions by stanbon, Cromlix:
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Where do lines y = x - 1 and y = -5x + 11 intersect?
-------
Substitute and solve for "x"::
x-1 = -5x+11
6x = 12
x = 2
----
Solve for "y"::
y = x - 1
y = 1
-----
Cheers,
Stan H.
------

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Hi there,
Solve as simultaneous equations by elimination.
y = x - 1......(1)
y = -5x + 11...(2)
Rearrange
y - x = -1.....(1)
y + 5x = 11....(2)
Multiply (1) by 5
5y - 5x = -5...(1)
y + 5x = 11...(20
6y.....= 6
y......= 1
Substitute y = 1 into Equatn(1)
y = x - 1
1 = x - 1
x = 2
The lines intersect at (2,1)
Hope this helps :-)

Question 993370: A javelin is thrown from a height of 2m above the ground.
It lands 60 meters away. The path of the javelin can be represented by the equation
a) What does x and y represent in this situation?
b)If it is known that the point (10,15) lies on the path given by the equation , use algebra and all the information given in a) and b) to show
that:
, and c = 2
c)Hence find the height of the javelin after travelling 20m horizontally
For a) Does y represent the height of the javelin, and x represent the horizontal distance traveled?
For b) I have no idea :c
For c) Really stuck!
It would be very helpful if you could draw up a graph too!! Thanks for help!!

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a) What does and represent in this situation?
represent height (dependent variable,depends on how changes )
represent the path (dependent variable,it is a variable that stands alone and isn't changed by the other variables you are trying to measure )
b)
If it is known that the point (,) lies on the path given by the equation , use algebra and all the information given in a) and b) to show that:
, and
if the point (,)=(,), we have

..............eq.1
if , and , we have equation like this:
...now,show that when

c)
Hence find the height of the javelin after traveling horizontally

Question 993056: (k - 2)x^2 + 50x + 2k + 3 = 0
How do I find the value of k if the roots are reciprocal
The ans is -5 but i don't know the method

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(k - 2)x^2 + 50x + 2k + 3 = 0
How do I find the value of k if the roots are reciprocal
The ans is -5 but i don't know the method
--------
Roots are:: [-50+-sqrt[50^2 - 4(k-2)*3]/2
----
If they are reciprocal::
[-50 + sqrt(50^2-12(k-2)]/2 = 1/[-50-sqrt(50^2-12(k-2)]/2
-----
Solve for "k"
=====
Cheers,
Stan H.
----------

Question 992750: 2[(x)/(x+2)]^2 - 3[(x)/(x+2)] - 9 = 0
Can you help me?

Found 2 solutions by AnlytcPhil, Edwin McCravy:
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Or you can use u-substitution:

Let

then the above becomes:

;

;

Then since

;

Mutiply both sides by x+2

;

;

;

;

Edwin

Answer by Edwin McCravy(13211)   (Show Source):
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Multiply through by (x+2)²

Divide through -2}}}

5x+6=0;   x+3=0
5x=-6     x=-3
x=-6/5

Edwin

Question 992751: 3x + 14 = 5
Can you help me solve this?

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3x + 14sqrt( x ) = 5
--------------------
3(sqrt(x))^2 + 14sqrt(x) - 5 = 0
-----
Let w = sqrt(x)
----
3w^2 + 14w - 5 = 0
---
Factor::
3w^2 + 15w-w -5 = 0
3w(w+5)-(w+5) = 0
(w+5)(3w-1) = 0
----
w = -5 or w = 1/3
----
Now, solve for "x"
sqrt(x) = -5 or sqrt(x) = 1/3
-----
x = 25 or is extraneous since sqrt(25) = 5, by definition; or x = 1/9
----------
Cheers,
Stan H.
-----------

Question 992752: 5x^(1/3) + 3x^(2/3) = 8
Can you help me? I hope I wrote it right; it's 5x to the one third plus 3x to the two thirds. Thank you!

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5x^(1/3) + 3x^(2/3) = 8
-----
Rewrite as::
3(x^(1/3))^2 + 5(x^(1/3)) - 8 = 0
-------
Note:: This is a quadratic equation. The variable is x^(1/3)
----
Let x^(1/3) = w
---
Rewrite the equation as::
3w^2 + 5w - 8 = 0
Factor::
3w^2 + 8w-3w - 8 = 0
w(3w+8) - (3w+8) = 0
(3w+8)(w-1) = 0
----
w = -8/3 or w = 1
----
Now, solve for "x"
x^(1/3) = (-8/3) or x^(1/3) = 1
----
x = (-8/3)^3 or x = 1^3
---
x = -512/27 or x = 1
-------------
Cheers,
Stan H.
------------

Question 992640: Write the expression (x-9)^2-4 in factored form and identify a,r, and s. assume r is less than or equal to s.
factor form= a(x-r)(x-s)

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Your given expression is the difference of squares.

Question 992637: Put the function 2x^2+12x+3 in vertex form and state the values of a,h, and k.
vertex form= a(x-h)^2+k

Question 992635: Put the equation y=4x^2+7x+6 in vertex form and state the values of a, h, and k. Enter the exact answers.
Vertex form= a(x-h)^2+k

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Put the equation y=4x^2+7x+6 in vertex form and state the values of a, h, and k. Enter the exact answers.
Vertex form= a(x-h)^2+k
------
y-6 = 4(x^2+(7/4)x)
-----
4(x^2+(7/4)x+(7/8)^2) = y - 6 +4(7/8)^2
-----
4(x+(7/8))^2 = y - 6 + 49/16
------
4(x+(7/8))^2 = y-(47/16)
-----
y = 4(x+(7/8))^2 + (47/16)
-------
a = 4
h = -7/8
k = 47/16
------------
Cheers,
Stan H.
-------------

Question 992634: Find the minimum value of x^2-2x-8, if it has one.
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Does the graph of: have a maximumum and minimum?
Well, it may have a maximum or it may have a minimum, but it doesn't have both!
The graph of a quadratic equation (), such as this, is called a "parabola" and a parabola has a maximum or a minimum depending upon whether it opens upwards (has a minimum) or opens downwards (has a maximum).
You can tell which way the parabola opens by inspecting the coefficient of the term, which is +1 in your problem.
If this coefficient is positive, then the parabola opens upwards and the graph has a minimum.
If the coefficient is negative,then the parabola opens downwards and the graph has a maximum.
The maximum/minimum point (also known as the "vertex") can be found as follows:
The x-coordinate of this point is:
The a is the coefficient of the term and the b is the coefficient of the term.

In your equation, and , so

This is the x-coordinate of the vertex.
To find the y-coordinate, substitute into the given quadratic equation and solve for :
Substitute .

The vertex (or the in this case) is at (, )
Let's see what the graph looks like:

Question 992497: 1/^x2 + 5/x + 6 = 0
Can you help me solve this equation?

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***********************************
is 1/^x2 really 1/(x^2)?, if yes then
***********************************
1/^x2 + 5/x + 6 = 0
multiply both sides of = by x^2
1 + 5x + 6x^2 = 0
rearrange terms
6x^2 + 5x + 1 = 0
factor polynomial
(3x + 1) * (2x + 1) = 0
the two solutions are
x = -1/3 and x = -1/2
**********************************
check the solutions
1) x = -1/3
(1 / (1/9)) + (5/(-1/3)) + 6 = 0
9 - 15 + 6 = 0
0 = 0
**********************************
2) x = -1/2
(1 / (1/4)) + (5/(-1/2)) + 6 = 0
4 -10 + 6 = 0
0 = 0

Question 992402: Let f(x) = 3x4 − 14x2 + 4x − 5.
Find the local maximum and minimum values of f and the values of x at which they occur.
Find the intervals on which f is increasing and on which f is decreasing.

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Let f(x) = 3x4 − 14x2 + 4x − 5.
Find the local maximum and minimum values of f and the values of x at which they occur.

Find the intervals on which f is increasing and on which f is decreasing.
f'(x) = 12x^3 - 28x + 4
---
Solve 12x^3 - 28x +4 = 0
3x^3 - 7x + 1 = 0
--------
x = -1.59 or 0.144 or 1.45
--------------
f"(x) = 36x^2 - 28
f"(-1.59) > 0 so min at x = -1.59
f"(0.144) < 0 so max at x = 0.144
f"(1.45) > 0 so min at x = 1.45
---------
Cheers,
Stan H.
----------------

Question 992364: Write an equation that has the given solutions. Thank you!!
i, −i

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.
It is well known equation

(x - i)*(x+i) = 0,

which is the same as

= .

Question 992365: Write an equation that has the given solutions. (There are many correct answers.)
8, , and

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.
= ,

which is the same as

= ,

for example.

Question 992362: Write an equation that has the given solutions.
-6/5 and 5/9

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an equation that has the given solutions and will be:

....multiply

Question 992241: A person standing close to the edge on the top of a 270-foot building throws a baseball vertically upward. The quadratic equation
models the ball's height \, h \, above the ground in feet, t seconds after it was thrown.
How high is the ball after 5 seconds?

??feet.
How many seconds does it take until the ball finally hits the ground? Round to the nearest tenth of a second.
?? seconds.
(I plugged in 5, then set it equal to 0, none of it worked, I'm probably doing it wrong)

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h = -16 t^2 + 160 t + 270
to find the height of the ball after 5 seconds substitute 5 for t
h = -16(5^2) + (160*5) + 270
h = 670 feet high after 5 seconds
*****************************************************************
to find when the ball hits the ground, set h = 0 and solve the quadratic for t
-16 t^2 + 160 t + 270 = 0
multiply both sides of = by -1
16t^2 - 160t - 270 = 0
use quadratic formula to solve for t
t = (160 + sqrt(160^2 - (4*16*(-270)))) / (2*16) = 11.471089553 approx 11.5
t = (160 - sqrt(160^2 - (4*16*(-270)))) / (2*16) = −1.471089553 approx -1.5
****************************************************************************
we want the positive value for t
t = 11.5 seconds for the ball to hit the ground

Question 992267: Construct the quadratic equation in the form az2+bz+c =0 having the following solutions. Find the coefficients a, band c for each equation.
(a)
(a) z=7, z=−2.
___ z2+ ___ z+ ___ =0

(b) z=−3+4i, z=−3−4i.
___ z2+ ___ z+ ___ =0

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You cannot construct THE quadratic equation knowing only two points on the curve, in this case the two zeros or x-intercepts. In the case of your first example, you know that one factor of the quadratic trinomial is and another factor is . But there could also be some real number such that the complete factorization of the trinomial would be and the family of quadratic equations related in that they would all have the same roots would be . Had the question asked you to construct A quadratic equation, that would have been a different story.

John

My calculator said it, I believe it, that settles it

Question 992292: y^2 - x^2 = 32 and y - x = 2 ...what's the value of x and y....thank you
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y^2 - x^2 = 32 and y - x = 2 ...what's the value of x and y....thank you
 ------ eq (i)
y = 2 + x -------- Solving 2nd equation for y ------ eq (ii)
1) Substitute 2 + x for y in eq (i)
2) Combine all like-terms and solve equation for x
3) Substitute value for x in eq (ii) to obtain value for y



Question 992229: A man built a walk of uniform width around a rectangular pool. If the area of the walk is 245 square feet and the dimensions of the pool are 6 feet by 22 feet, how wide is the walk?
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Same as #992205, which already was given a response.

Question 992195: A rectangle is 6 times as long as it is wide. The area is 96 square feet. What is the perimeter of the rectangle?

Found 2 solutions by stanbon, Alan3354:
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A rectangle is 6 times as long as it is wide. The area is 96 square feet. What is the perimeter of the rectangle?
---------
width:: x
length:: 6x
----
Equation:
x*6x = 96
----
6x^2 = 96
----
x^2 = 16
---
x = 4 (width)
6x = 24 (length)
---------------
Perimeter = 2(length+width)
P = 2(24+4)
P = 2(28)
P = 56
-----------
Cheers,
Stan H.
-----------

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A rectangle is 6 times as long as it is wide. The area is 96 square feet. What is the perimeter of the rectangle?
-----
L = 6W
L*W = 96
---
6W*W = 96
W*W = 16
W = 4 ft
------------
Perimeter = 2W + 2L

Question 992196: Solve the following equation. Separate your answers with commas. Repeated roots should only be entered once.
1/x-7 + 1/x-2 = 9/14

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Solve the following equation. Separate your answers with commas. Repeated roots should only be entered once.
1/x-7 + 1/x-2 = 9/14
------------------------
Multiply thru by 14(x-2)(x-7) to get:
14(x-2) + 14(x-7) = (x-2)(x-7)
------
28x -126 = x^2 - 9x + 14
------
x^2 - 37x + 140 = 0
-------------
x = 32.72... or x = 4.28...
=============
Cheers,
Stan H.
-------------

Question 992194: A person standing close to the edge on the top of a 270-foot building throws a baseball vertically upward. The quadratic equation
h = -16 t^2 + 160 t + 270
models the ball's height \, h \, above the ground in feet, t seconds after it was thrown.
How high is the ball after 5 seconds?

How many seconds does it take until the ball finally hits the ground? Round to the nearest tenth of a second.

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A person standing close to the edge on the top of a 270-foot building throws a baseball vertically upward. The quadratic equation
h = -16 t^2 + 160 t + 270
models the ball's height h above the ground in feet, t seconds after it was thrown.
How high is the ball after 5 seconds?
----
h(t) = -16t^2 + 160t + 270
Find h(5) = sub 5 for t.
h(5) is 670 feet.
---------------------
How many seconds does it take until the ball finally hits the ground? Round to the nearest tenth of a second.
h(t) = -16t^2 + 160t + 270
-16t^2 + 160t + 270 = 0
Solve for t. Use the positive solution.
-----
 Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc) Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=42880 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: -1.47108955277239, 11.4710895527724. Here's your graph:

==================================================
t = 11.5 seconds.
Maybe you didn't use the nearest 10th ?

Find the whole number such that twice its square added to itself makes 10.

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.
x = the number
.

Since we are asked for a whole number:
x=2
.

Question 991899: 1+2+3=40
2+3+4=85
3+4+5=148
4+5+6=229
5+6+7=???