Questions on Algebra: Quadratic Equation answered by real tutors!

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Y=mX+b (m=slope).
5y+8x+4=10+8x
5y=-4+10-8x+8x
5y=6
y=6/5 slope=0.
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, y =6/5).

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X-Intercept(s)

Start with the given function


Plug in


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


Break up the fraction.


Reduce.


or Break up the "plus/minus"


So the answers are or


which approximate to or


So the x-intercepts are and which in decimal form is (6.236,0) and (1.764)

---

Y-Intercept

Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the y-intercept is (0,11)

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a)

Start with the given equation


Divide both sides by 2


or Take the square root of both sides.


So the solutions are or


which approximate to or


==================================================

b)

Start with the given function


Plug in


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


Break up the fraction.


Reduce.


or Break up the "plus/minus" to form two equations.


So the answers are or


which approximate to or


So this means that the x-intercepts are and

which in decimal form are (1.732,0) and (-1.732,0)


======================================================


Notice how the solutions of the first equation are simply the x-coordinates of the x-intercepts.

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Simply let "x" be any number (since "x" isn't even listed), but y must be equal to -6.


So we could have the points (-5,-6), (-4,-6), (-3,-6), (-2,-6), (-1,-6), (0,-6), (1,-6), (2,-6), (3,-6), (4,-6), (5,-6). Notice how these points have the y-coordinate of -6



Now plot these points:






Now draw a line through all of these points to graph (notice how it's a horizontal line)



Graph of through the given points.

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Remember: For rectangle,
----------------------------> working eqn
Given:



Substitute,
------>
BY QUADRATIC: , , &




---> 2 values ,
USED ----------------> LENGTH
-----------------> WIDTH
In doubt? Go back working eqn:


Thank you,
Jojo

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Solve by completing the square
m^2-1/2m=7/2
----------------
m^2 -(1/2)m + (1/4)^2 = (7/2) + (1/4)^2
(m - (1/4))^2 = 56/16+1/16
(m - (1/4)) = +-(1/4)sqrt(57)
m = (1/4)+(1/4)sqrt(57) or m = (1/4)-(1/4)sqrt(57)
=========================================================================
Cheers,
Stan H.

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X+X+2+X+4+X+6=216
4X+12=216
4X=216-12
4X=204
X=204/4
X=51 FOR THE LOWEST INTEGER.
51+2=53 FOR THE NEXT INTEGER.
51+4=55 FOR THE THIRD INTEGER
51+6=57 FOR THE LARGEST INTEGER.
PROOF:
51+53+55+57=216
216=216

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the sum of four consecutive odd integers 216.find the four integers.
------------------------------------
1st 2x-3
2nd 2x-1
3rd 2x+1
4th 2x+3
------------------
EQUATION:
8x = 216
x = 27
------------
1st : 2x-3 = 54-3 = 51
2nd: 53
3rd: 55
4th: 57
==================
Cheers,
Stan H.

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Fnd three consecutive integers whose sum is 126.
--------------------
The average will be 126/3, which is 42. That's the middle number of the 3, so it's 41, 42, 43.

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The line equation is: Y=mX+b where m=slope
x+11=y
y=x+11 has a slope=1.
y-x=-7
y=x-7 has a slope=1
Therefore these 2 lines are parallel.

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The line equation is: Y=mX+b where m=slope
x+11=y
y=x+11 has a slope=1.
y-x=-7
y=x-7 has a slope=1
Therefore these 2 lines are parallel.

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(y^2-4w^2)^2 = (y^2)^2 - 2(y^2)(4w^2) + (4w^2)^2
= y^4 - 8y^2w^2 + 16w^4
=============================
Cheers,
Stan H.

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(y^2-4w^2)^2
y^2-4w^2
y^2-4w^2 Now multiply
-------------------------------
y^4-4y^2w^2-4y^2w^2+16w^4
y^4-8y^2w^2+16w^2 Answer.

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We have two sides a and b (length,width) but because b is just twice the a, we can express length of both sides using only one variable: a, 2a
after plugging in all the variables you should be left with quadratic equation in one variable.
Cheers,
Adam
p.s. If you don't know which equation should you plug into, think about one which contains both sides of rectangle and it's perimeter...

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x=length
y=width
the length of a rectangle is twice the width
then x=2y
the perimeter is 48 in.
then 2(x+y)=48 then x+y=24


then then

Answer: length is 16 in and width is 8 in

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Start with the given equation


Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:

or


or Now solve for x in each case


So the answers are


or

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(3, )
just put t=3 and:
if t =3 then
so the pair is (3,0)

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Let x = width


Since the "height is 4 feet more than its width", this means that the height is feet.

Start with Pythagoreans Theorem. Note: "a" and "b" are the legs of the triangle (in this case the width and height of the door) and "c" is the hypotenuse (which in this problem is the diagonal of the door)


Plug in (this is the width), (this is the height), and (which is given as the diagonal)


Square to get


Reduce


FOIL


Combine like terms.


Multiply every term by the LCD 2 to clear the fraction


Subtract 97 from both sides.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible widths are or (which in decimal form are or respectively)


However, since a negative width doesn't make sense, this means that the only solution is (which is the mixed fraction )


So the width is 2 and a half feet


Go back to the expression that represents the height


Plug in )


Add


So the height of the door is 6 and a half feet.


-------------------------------------
Answer:
So the dimensions of the door are feet by which means that the answer is B) 2 1/2 ft by 6 1/2 ft

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Solve for x:
Add to both sides.
Square both sides..
Subtract 169x from both sides.
Solve the quadratic equation by factoring.
Apply the zero product property: If a*b = 0, then either a = 0 or b = 0 or both.
or so...
or

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x^3 + 6x^2-7x-60,(x-3) and (x+4)
-------------
X-3
X+4 MULTIPLY
----------
X^2-3X+4X-12=X^2+X-12
NOW DIVIDE THE X^3 EQUATION BY THIS PRODUCT TO GET THE THIRD FACTOR:
------------------X+5 ANSWER FOR THE THIRD FACTOR.
---------_________________
X^2+X-12 )X^3+6X^2-7X-60
**********X^3+X^2-12X
**********-------------
**************5X^2+5X-60
**************5X^2+5X-60
**************-----------
**************0+0+0
PROOF:
X^2+X-12
X+5 MULTIPLY
----------
X^3+X^2-12X+5X^2+5X-60
X^3+6X^2-7X-60

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32y^2 - 12y - 2
2(16y^2-6y-1)
2(8y+1)(2y-1)
------------------
36x^2 + 36x + 9
9(4x^2-4x+1)
9(2x-1)(2x-1)
---------------------
3x^2 + 54x + 195
3(x^2+18x+65)
3(x+13)(x+5)

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Remember, not every trinomial is factorable. If not factorable, they are called "prime".
Factor:
Look at the constant term, 51, its factors are 3 and 17.
Now, can you ever get the middle coefficient of -16 by adding or subtracting 3 and 17?
3+17 = 20
-3+(-17) = -20
3-17 = -14
-3-(-17) = 14
The answer is no!
So the trinomial is, therefore, prime!

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Solve for x:
Simplify the left side by factoring an x.
Simplify.
Apply the zero product rule: If a*b = 0, the either a =0, or b = 0, or both.
or so
Solution:

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Think about it this way.

If , then and .


If , then and .


If , then and .


If , then and .



So we have this table of values

x	f(x)	g(x)
0	1	0
1	3	1
2	9	2
3	27	3

From the table, we can see that increments by 1 as x increments by 1. On the other hand, we can see that goes from 1 to 3 (a difference of 2), 3 to 9 (a difference of 6), 9 to 27 (a difference of 18), etc. So the differences between each term is: 2, 6, 18, etc....


This means that from x=0 to x=1, the average rate of change for g(x) is 2. From x=1 to x=2, the average rate of change for g(x) is 6. From x=2 to x=3, the average rate of change for f(x) is 18.

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

So dividing the first average rate of change 2 by 1, we get . So from x=0 to x=1, is growing twice as fast as .


Dividing the second average rate of change 6 by 1, we get . So from x=1 to x=2, is growing six times as fast as .


Dividing the third average rate of change 18 by 1, we get . So from x=2 to x=3, is growing eighteen times as fast as .


As you can see, the exponential function is not growing at a constant rate. So cannot be growing 3 times faster than




Note: the function does however grow three times faster than , but that is for another problem.



So that means that the statement is false.

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# 4

Let x = speed of boat in still water



Start with the distance-rate-time formula



Plug in and . This equation represents the upstream journey


Divide both sides by to isolate "t"


So the expression for the time it takes to go upstream can be represented by the expression

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

Plug in and . This equation represents the downstream journey


Divide both sides by to isolate "t"


So the expression for the time it takes to go downstream can be represented by the expression


Now simply add the two time expressions to get:


Now set that expression equal to the total time of 9 hours


Multiply every term by the LCD to clear the denominators


FOIL


Distribute


Subtract from both sides. Add to both sides.


Combine like terms


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


or Break up the expression.


So the answers are or


which approximate to or


Since a negative speed doesn't make sense in this problem, this means that the only solution is

--------------------------------------------------------------------------------
Answer:

So the speed of the boat in still water is approximately 13.98 mph (rounded to the nearest hundredth).

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# 3

Let x = original speed (ie slower speed)


Start with the distance-rate-time formula


Plug in and


Divide both sides by "x" to isolate "t"


The statement "if he had gone 10mph faster, the trip would have taken 1 hour less" tells us that the new speed is and the new time is


Go back to the distance-rate-time formula


Plug in , , and replace with


FOIL


Plug in


Multiply


Multiply every term by the LCD "x" to clear the denominator


Subtract 200x from both sides


Combine like terms


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible answers are or

Since a negative speed doesn't make sense, this means that the only solution is

-------------------------------------------------------------------------------
Answer:

So the original speed was 40 mph

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# 2

Start with the given formula


Plug in


Distribute


So at the end of the first year, he has dollars in the account


Since "At the beginning of the second year, an additional $3500 is invested", this means that we simply add 3,500 to the amount to get


So at the beginning of the second year, he invests dollars


So this time


Go back to the given formula


Plug in (this is the amount that is in the account after the second year) and


FOIL


Subtract 15,569.75 from both sides


Combine like terms


Multiply every term by the 100 to clear the decimals.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for r


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible answers are or

which approximate to or


However, since a negative interest rate doesn't make much sense, this means that the only solution is which is the percentage 8.5%


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

Answer:
So the interest rate is 8.5%

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# 1

Let x = width of path

First let's find the area of the garden only: Area of Garden = 18*13=234

So the area of the garden only is 234 square feet.

Now let's draw a picture:



From the picture, notice that the length of the entire rectangle (including the width of the path) is (notice there are two "x" lengths per side) and the total width is .


So the area of the entire enclosure (including the path) is the expression


Now subtract off the area of the garden (we only want the area of the path)


FOIL


Combine like terms


So the area of the path only is


Start with the area of the path


Plug in (which is the area of the path)


Subtract 516 from both sides


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or

Since a negative width doesn't make sense, this means that the only solution is


-------------------------------------------------------------
Answer:

So the width of the path can be up to 6 feet (ie 6 feet is the maximum width)

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1.m(7-2m)=6
7m - 2m^2 = 6
2m^2-7m+6 = 0
2m^2-3m-4m+6 = 0
m(2m-3)-2(2m-3) = 0
(2m-3)(m-2) = 0
m = 3/2 or m=2
--------------------------
2. 1/x = (x-2)/24
24x = x-2
23x = -2
x = -2/23
--------------------------
3. 4/(x+10)=4/(x-1)
1/(x+10) = 1/(x-1)
x-1 = x+10
no solution
Your posting need parentheses.
-------------------
4.x+3/x=x-4/3
Needs parentheses.
5.40/x-20/x-3=8/7
need parentheses
----------------------
Cheers,
Stan H.

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y = x^2-2x+1
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, x^2 -2x +1).
Turning point (1,0).

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complete the square.
y=x^2-4x+2
-------------
x^2-4x = -y-2
x^2-4x+4 = -y-2+4
(x-2)^2 = -(y-2)
=====================
Cheers,
Stan H.

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solve by completing the square
2x^2+5x-1=0
2x^2 + 5x = 1
2(x^2 + (5/2)x + (5/4)^2) = 1 + 2(5/4)^2
2(x+(5/4))^2 = 1 + 2(25/16) = 1 + (25/8) = 33/8
(x+(5/4))^2 = 33/16
Take the square root:
x+(5/4) = [sqrt33]/4 or x+(5/4) = -[sqrt33]/4
x = [-5+sqrt(33)]/4 or x = [-5-sqrt(33)]/4
============================================
Cheers,
Stan H.

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x(x+5)=-10
x^2+5x+10=0

x=(-5+-sqrt[5^2-4*1*10])/2*1
x=(-5+-sqrt[25-40])/2
x=(-5+-sqrt-15)/2
x=(-5+-3.873)/2
x=-2.5+-1.9365i answer.

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Solve by completing the square:
First, divide through by 2 to get the coefficient of equal to 1.
Now add to both sides.
Complete the square in the x-terms by add the square of half the x-coefficient () to both sides of the equation.
Simplify, where possible.
Now factor the left side.
Simplify.
Take the square root of both sides.
Simplify.
Now subtract from both sides.

or

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a)

It will touch the x-axis when the polynomial has only one solution. If the polynomial has only one solution, then the discriminant is equal to zero.


Start with the discriminant formula


Plug in , , and


Square -3 to get 9


Multiply


Subtract 9 from both sides


Divide both sides by -4


So when , then the polynomial will have only one solution and will touch the x-axis.

---

b)

Follow the same procedure described above, but this time .

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Hazel has a screen door whose height is 4 feet more than its width. She wishes to stabilize the door by attaching a steel cable diagonally. If the cable measures (sqrt 19)/2 ft, what are the dimensions of the door? (1 point)
A) 2 1/4 ft by 6 1/4 ft
B) 2 1/2 ft by 6 1/2 ft
C) 3 ft by 7 ft
D) 3 1/2 ft by 7 1/2 ft
--------------------------------
Draw the picture.
hypotenuse = [sqrt(19)]/2 ft.
Width : x ft.
height: x+4 ft.
-------------------
EQUATION:
x^2 + (x+4)^2 = [sqrt(19)/2]
x^2 + x^2 + 8x + 16 = 19/4
2x^2 + 8x + 11.25 = 0
x = [-8 +- sqrt(64 - 4*2*11.25)]/4
x = [-8 +- sqrt(-26)]4
Note: This value is not a Real Number.
------------------------------
Something is wrong in the statement of your problem.
-----------------------------
Cheers,
Stan H.

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find a quadratic equation where f(2)=16 and the minimum value of f is f(-2)=-16
----------------------------------
Plot those points and you will see that f(-6) = 16 by symmetry about
the x-axis
-----------------
So you have three points: (-6,16),(-2,-16),(2,16)
----------------------------------------------------
Substitute these into ax^2 + bx + c = y
to get three equations with unknowns a,b, and c.
36a -6b + c = 16
4a -2b + c =-16
4a +2b + c = 16
-------------------------
Use any method to solve this system to get:
a = 2
b = 8
c =-8
---------------
EQUATION:
y = 2x^2 + 8x -8
==========================
Cheers,
Stan H.

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n^2-9n+14=0
(N-7)(N-2)=0
N-7=0
N=7 ANSWER.
N-2=0
N=2 ANSWER.

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Start with the given equation


Subtract 7 from both sides

Solved by pluggable solver: Computing the Discriminant

From we can see that , , and Start with the discriminant formula. Plug in , , and Square to get Multiply to get Rewrite as Add to to get Since the discriminant is greater than zero, this means that there are two real solutions.

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x^2+10x-8=0
then

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Start with the discriminant formula


Plug in , and