Why is it impossible to root a whole equation

Question

I have this equation:

$x^2+(x-7)^2=13^2$

Why can't squaring the whole equation be used to solve it?

$\sqrt{x^2+(x-7)^2=13^2}\equiv x+x-7=13$

My question is merely asking when square cannot be used to solve an equation.

Answer 1

Following your approach, we would get $x=10$.

But neither $10^2+3^2=13^2$, nor $\sqrt{10^2+3^2}=13$.

You can indeed "square root" the equation (with some care for the signs), but you may not "square root" a sum.

Indeed, $(a+b)^2=a^2+2ab+b^2\ne a^2+b^2$ so that $a+b\ne\sqrt{a^2+b^2}$.