solve the absolute value equation
solve the absolute value equation
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.