Solving $\sqrt{25 - 10x + x^2} = x-5$

Question

I need help solving this math question.

$$\sqrt{25-10x+x^2}=x-5$$

I got $x=5$, but apparently it is wrong.

Please provide an explanation if possible. Thanks.

Answer 1

This is an identity... or kinda. You see $\sqrt{x^2-10x+25}=\sqrt{(x-5)^2}=|x-5|$. So, if you are thinking that $x\geq 5$, then, you'll have $\sqrt{x^2-10x+25}=x-5$. So, this is not an equation.

Answer 2

HINT: $\sqrt{25-10x+x^2}=\sqrt{(5-x)^2}=|5-x|=|x-5|$. Thus, your equation can be rewritten as

$$|x-5|=x-5\;.$$

For what values of $x$ is that true? (Think about the definition of the absolute value.)

Answer 3

Hint: The left hand side can be rewritten as: $$ \sqrt{(x - 5)^2} = |x - 5| $$ Do you know how to rewrite absolute values as piecewise functions? Try that. Note that there are two cases to consider: when $x < 5$ and when $x \geq 5$. Note that there are solutions other than $5$; for example, $x = 7$ is a solution (plugging it in to both sides yields $2 = 2$).