quadratics - My brother asked me to explain a algebra problem. How should I explain it?

So the problem is:

$$\frac{4}{x}+\frac{6}{2}=x$$
And we solved it using the pq formula. But than he asked me:

How do I know when I should apply pq to similar equations like this
and not just:

$$4 + 3 = x^2$$

$$x = \sqrt{7}$$ ?

Do I have to just test to find out if its wrong, and than just try all
possible solutions until I find the correct one or can I see it some
how?

I saw right away how to solve it but I do not really know why, so long ago I dealt with problems like this. Anyone have a idea how to explain it?

Answer

It may help if you write your equation as you might on paper:

$$\frac{4}{x}+\frac{6}{2}=x.$$ Then, supposing $x\neq 0$, you can see that to remove the $x$ in the denominator of the first fraction you should multiply all top terms by $x$ to get $$\frac{4\times x}{x}+\frac{6\times x}{2}=x\times x.$$ Now we can simplify to get $$4+3x=x^2.$$ Basically you will need to learn the rules of algebra and that can often be helped by writing things out "nicely". Rearranging you then get $x^2-3x-4=0$ which you can solve using the "pq" formula.