How to solve |x^2 - 5x + 4| = 4? (Why is my solution wrong?)

Question

zyberg · Answer

My solution, that gives off too many roots:
First I split this into two cases:
1) x^2 - 5x + 4 = 4
2) -(x^2 - 5x + 4) = 4; x^2 - 5x + 4 = -4

Solving 1:
x(x - 5) = 0; x = 0 or x = 5.

Solving 2:
x^2 - 5x + 8 = 0;
x = 2 or 3.

The answers in my book say that it's only 0, 5 (so, only the first case is the right one). Why is that?

solomonzelman · Answer

$x^2 - 5x + (2.5/2)^2 = -8+ (2.5/2)^2$

solomonzelman · Answer

$x^2 - 5x + (2.5/2)^2 = -8+ (2.5/2)^2$
 $x^2 - 5x + (6.25/4) = -8+ (6.25/4)$
 $(x-2.5)^2 =(-32+6.25)/4<0$

See what I am doing?

solomonzelman · Answer

so, your solutions are clearly imaginary numbers.
(No real solutions.)

zyberg · Answer

Thank you very much! :)

solomonzelman · Answer

No problem:)