What's in a squared?

Consider 3 case:

(i) if $x=y=0,$ we can have $x^2=y^2=0.$ It means statement $x^2=y^2 \implies x=y$ always true.

(ii) if $x<0,y>0$ ( or $x>0,y<0$ ), we can have $x^2>0,y^2>0.$ It means statement $x^2=y^2 \implies x=y$ always false, because the signs $x,y$ are opposite.

(iii) if $x<0,y<0$ ( or $x>0, y>0$ ), we can have $x^2>0,y^2>0.$ It means statement $x^2=y^2 \implies x=y$ always true, because the sign $x,y$ are same.

From (i),(ii) and (iii) $\boxed{sometimes true, sometimes false}$

If $x^2=a \Rightarrow x$ could be positive or negative.

If $y^2=a (x^2=y^2) \Rightarrow y$ could be positive or negative.

Therefore, it is true and false.

$x^2$ = $y^2$ means $x^2$ - $y^2$ = 0 so either x-y=0 (x=y) or x+y=0 (x=-y) so x and y are not always equal

1. x^2 =y^2 ; 2. x^2-y^2 = 0; 3. Factor: (x+y) (x-y) =0
2. So, its is either x+y = o or x-y =o 5. Finally It is either x=-y or x=y.

the square root of x and y could be either positive or negative...

for example: 4=4

both x^2 and y^2 is 4

4 could have a square root of +2 and -2

if that's the case, +2 could be x and -2 could be y, which in that case, x=y won't be true

square root results in both positive and negative signs......... so not true always ....

in general , if x^2=y^2 . then x\quad =\quad \pm \sqrt { { y }^{ 2 } } therefore ,, x\quad =\quad \pm y

Sometimes true, sometimes false. √x²=√y², x=y, or, x=(-y), or, (-x)=y.

Counter example.

x = 2 and y = -2