How do you solve $｜x-2｜ =｜x^2-4｜$ ?

Question

1426 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-10T06:31:09

$x = -1$ , $x = -3$ or $x = 2$

Given:

$abs(x-2) = abs(x^2-4)$

We must have one of the following:

a) $color(white)(0)(x-2) = (x^2-4)$

b) $color(white)(0)(x-2) = -(x^2-4)$

$color(white)()$
Case a)

$x-2=x^2-4$

Subtract $(x-2)$ from both sides to get:

$0 = x^2-x-2 = (x-2)(x+1)$

So $x=-1$ or $x=2$

$color(white)()$
Case b)

$x-2=-(x^2-4)$

Add $(x^2-4)$ to both sides and transpose to get:

$0 = x^2+x-6 = (x+3)(x-2)$

So $x=-3$ or $x=2$

$color(white)()$
Check potential solutions

Trying each of these values of $x$ as possible solutions of the original equation:

$abs((-1)-2) = abs(-3) = abs((-1)^2-4)$

$abs((2)-2) = 0 = abs((2)^2-4)$

$abs((-3)-2) = 5 = abs((-3)^2-4)$

So all the possible solutions are solutions of the original equation.

How do you solve ｜x-2｜ =｜x^2-4｜｜x-2｜ =｜x^2-4｜?