Rewriting an algebraic equation with square roots

Question

In as part of solving a question, the equation

$a-3\sqrt a-4=0$

is written into

$a^2-3a-4=0$

How is this done? Do you square everything in the equation? But in this case why are only the $a$ squared?

Thanks!

Answer 1

This is a trick, and you are probably confused that they changed the variable, but did not change the variable name.

I change the variable name, so the trick is to define $b=\sqrt a$.

Then $b^2 - 3b - 4 = 0$.

This is exactly your second equation, but where the $a$'s have been replaced by $b$'s. From your first equation $a-3\sqrt a - 4 =0$, when $b=\sqrt a$, you get $a = b^2$, so this is why the equation becomes $b^2-3b-4=0$.