solve the absolute value equation

solve the absolute value equation

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x = 0 is obviously a solution.

If x ≠ 0, divide both sides of |x^2-2x| = |x^2+6x| by x,

|x-2| = |x+6|

Therefore are two ways to break the absolute signs:

x-2 = x+6, which has no solutions,

or

x-2 = -(x+6) => x = -2

Answer: x = 0, -2

Initially, there appears to be four possibilities from this:

x^{2} - 2x = x^{2} + 6x

x^{2} - 2x = -(x^{2} + 6x)

-(x^{2} - 2x) = -(x^{2} + 6x)

-(x^{2} - 2x) = x^{2} + 6x

After further examination, multiplying the first equation by -1 produces the third equation and multiplying the second equation by -1 produces the fourth equation. We only need to solve two problems:

x^{2} - 2x = x^{2} + 6x and x^{2} - 2x = -(x^{2} + 6x)

For the first possibility:

x^{2} - 2x = x^{2} + 6x Equation found when absolute value signs removed

x^{2} - 2x - x^{2} = x^{2} + 6x - x^{2} Subtract x^{2} from each side

-2x = 6x Simplify

-2x - 6x = 6x - 6x Subtract 6x from each side

-8x = 0 Simplify

-8x/(-8) = 0/(-8) Divide each side by -8

x = 0 Simplify

For the second possibility:

x^{2} - 2x = -(x^{2} + 6x) Second equation found when removing absolute value signs

x^{2} - 2x = -x^{2} - 6x Distribute the negative

x^{2} - 2x + x^{2} = -x^{2} - 6x + x^{2} Add x^{2} to both sides

2x^{2} - 2x = -6x Simplify

2x^{2} - 2x + 6x = -6x + 6x Add 6x to both sides

2x^{2} + 4x = 0 Simplify

2x(x + 2) = 0 Factor the left side

2x = 0 x + 2 = 0 Set each factor equal to 0

x = 0 x = -2 Solve each factor for x

The possible answers are 0 and -2. x = {-2,0}

To check the problem, lets plug the answers into the originals and see if they are correct.

l(-2)^{2} - 2(-2)l = l(-2)^{2} + 6(-2)l

l(4 + 4)l = l4 - 12l

l8l = l-8l

8 = 8

-2 checked as an answer

l0^{2} - 2*0l = l0^{2} + 6*0l

l0-0l = l0 -0l

l0l = l0l

0 = 0

0 checked as an answer.