# Find the Quadratic Constant of Variation x=4y^2+8y-3

x=4y2+8y-3
Rewrite the equation as 4y2+8y-3=x.
4y2+8y-3=x
Move x to the left side of the equation by subtracting it from both sides.
4y2+8y-3-x=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=4, b=8, and c=-3-x into the quadratic formula and solve for y.
-8±82-4⋅(4⋅(-3-x))2⋅4
Simplify.
Simplify the numerator.
Raise 8 to the power of 2.
y=-8±64-4⋅(4⋅(-3-x))2⋅4
Apply the distributive property.
y=-8±64-4⋅(4⋅-3+4(-x))2⋅4
Multiply 4 by -3.
y=-8±64-4⋅(-12+4(-x))2⋅4
Multiply -1 by 4.
y=-8±64-4⋅(-12-4x)2⋅4
Apply the distributive property.
y=-8±64-4⋅-12-4(-4x)2⋅4
Multiply -4 by -12.
y=-8±64+48-4(-4x)2⋅4
Multiply -4 by -4.
y=-8±64+48+16×2⋅4
y=-8±112+16×2⋅4
Factor 16 out of 112+16x.
Factor 16 out of 112.
y=-8±16⋅7+16×2⋅4
Factor 16 out of 16⋅7+16x.
y=-8±16(7+x)2⋅4
y=-8±16(7+x)2⋅4
Rewrite 16(7+x) as (22)2(7+x).
Rewrite 16 as 42.
y=-8±42(7+x)2⋅4
Rewrite 4 as 22.
y=-8±(22)2(7+x)2⋅4
y=-8±(22)2(7+x)2⋅4
Pull terms out from under the radical.
y=-8±227+x2⋅4
Raise 2 to the power of 2.
y=-8±47+x2⋅4
y=-8±47+x2⋅4
Multiply 2 by 4.
y=-8±47+x8
Simplify -8±47+x8.
y=-2±7+x2
y=-2±7+x2
Simplify the expression to solve for the + portion of the ±.
Simplify the numerator.
Raise 8 to the power of 2.
y=-8±64-4⋅(4⋅(-3-x))2⋅4
Apply the distributive property.
y=-8±64-4⋅(4⋅-3+4(-x))2⋅4
Multiply 4 by -3.
y=-8±64-4⋅(-12+4(-x))2⋅4
Multiply -1 by 4.
y=-8±64-4⋅(-12-4x)2⋅4
Apply the distributive property.
y=-8±64-4⋅-12-4(-4x)2⋅4
Multiply -4 by -12.
y=-8±64+48-4(-4x)2⋅4
Multiply -4 by -4.
y=-8±64+48+16×2⋅4
y=-8±112+16×2⋅4
Factor 16 out of 112+16x.
Factor 16 out of 112.
y=-8±16⋅7+16×2⋅4
Factor 16 out of 16⋅7+16x.
y=-8±16(7+x)2⋅4
y=-8±16(7+x)2⋅4
Rewrite 16(7+x) as (22)2(7+x).
Rewrite 16 as 42.
y=-8±42(7+x)2⋅4
Rewrite 4 as 22.
y=-8±(22)2(7+x)2⋅4
y=-8±(22)2(7+x)2⋅4
Pull terms out from under the radical.
y=-8±227+x2⋅4
Raise 2 to the power of 2.
y=-8±47+x2⋅4
y=-8±47+x2⋅4
Multiply 2 by 4.
y=-8±47+x8
Simplify -8±47+x8.
y=-2±7+x2
Change the ± to +.
y=-2+7+x2
Rewrite -2 as -1(2).
y=-1⋅2+7+x2
Factor -1 out of 7+x.
y=-1⋅2-1(-7+x)2
Factor -1 out of -1(2)-1(-7+x).
y=-1(2-7+x)2
Move the negative in front of the fraction.
y=-2-7+x2
y=-2-7+x2
Simplify the expression to solve for the – portion of the ±.
Simplify the numerator.
Raise 8 to the power of 2.
y=-8±64-4⋅(4⋅(-3-x))2⋅4
Apply the distributive property.
y=-8±64-4⋅(4⋅-3+4(-x))2⋅4
Multiply 4 by -3.
y=-8±64-4⋅(-12+4(-x))2⋅4
Multiply -1 by 4.
y=-8±64-4⋅(-12-4x)2⋅4
Apply the distributive property.
y=-8±64-4⋅-12-4(-4x)2⋅4
Multiply -4 by -12.
y=-8±64+48-4(-4x)2⋅4
Multiply -4 by -4.
y=-8±64+48+16×2⋅4
y=-8±112+16×2⋅4
Factor 16 out of 112+16x.
Factor 16 out of 112.
y=-8±16⋅7+16×2⋅4
Factor 16 out of 16⋅7+16x.
y=-8±16(7+x)2⋅4
y=-8±16(7+x)2⋅4
Rewrite 16(7+x) as (22)2(7+x).
Rewrite 16 as 42.
y=-8±42(7+x)2⋅4
Rewrite 4 as 22.
y=-8±(22)2(7+x)2⋅4
y=-8±(22)2(7+x)2⋅4
Pull terms out from under the radical.
y=-8±227+x2⋅4
Raise 2 to the power of 2.
y=-8±47+x2⋅4
y=-8±47+x2⋅4
Multiply 2 by 4.
y=-8±47+x8
Simplify -8±47+x8.
y=-2±7+x2
Change the ± to -.
y=-2-7+x2
Factor -1 out of -2-7+x.
Reorder 7 and x.
y=-2-x+72
Rewrite -2 as -1(2).
y=-1⋅2-x+72
Factor -1 out of -x+7.
y=-1⋅2-(x+7)2
Factor -1 out of -1(2)-(x+7).
y=-1(2+x+7)2
Rewrite -1(2+x+7) as -(2+x+7).
y=-(2+x+7)2
y=-(2+x+7)2
Move the negative in front of the fraction.
y=-2+x+72
y=-2+x+72
The final answer is the combination of both solutions.
y=-2-7+x2
y=-2+x+72
The given equation y=-2-7+x2,-2+x+72 can not be written as y=kx2, so y doesn’t vary directly with x2.
y doesn’t vary directly with x
Find the Quadratic Constant of Variation x=4y^2+8y-3