Proof that if: a^2=b^2 then a=b or a=-b

Question

I have to prove that if a^2=b^2 then a=b or a=-b, and I came up with the following and I am wondering if it is valid use of absolute values?: a^2=b^2 |square root of both sides

|a|=|b| |particularly interested in whether or not this next step is valid:

a=b or a=-b

Thank you in advance.

Edit: I do understand that perhaps easier way to prove is the following: a^2=b^2

a^2-b^2=0

(a+b)=0 or (a-b)=0

a=-b or a=b

But I am interested in whether or not my prior proof was valid.

Answer 1

That is correct, from the logical point of view. But it is incomplete. You proved correctly that, if $a^2=b^2$, then $|a|=|b|$. But now you still have to prove that if $|a|=|b|$, then $a=b$ or $a=-b$.

Answer 2

$a^2=b^2$ means $a^2-b^2 = 0$ and so $(a-b)(a+b)=0$. Thus $a=b$ or $a=-b$.

Answer 3

Overkill

Consider the quadratic equation

$x^2-b^2=0;$

Two real solutions

$x_{1,2}=\sqrt{b^2}=\pm|b|;$

$x_1=|b|;$ $x_2=-|b|;$ which for

1)$b\ge 0:$ $x_1=b;$ $x_2 =-b;$

2)$b< 0:$ $x_1=-b;$ $x_2=b;$ or

with the original $a:$ $a=\pm b;$