Question

Why is this correct? Why am I allowed to simplify just the top of this faction and leave the bottom? I thought whatever you do to the top you must do to the bottom

Answer 1

There's no reason why the 5 at the right hand side becomes 25 in the second line. Also, you're missing the $\pm$ sign when taking the square root of both sides. Here are the correct steps.

Start with

$x^2+x^2=5$

Add like terms:

$2x^2=5$

Divide both sides by 2:

$x^2=\dfrac{5}{2}$

Take the square root of both sides, considering both signs for the right hand side:

$x=\pm\sqrt{\dfrac{5}{2}}$