Are we allowed to multiply a denominator that has a variable with zero?

Question

I was given a question that resulted in the following equation:

$$ \cos(90°) = \frac{-10 + p}{\sqrt{11} \cdot \sqrt{20+p}}$$ Since $\cos(90°) = 0$, the equation becomes: $$ 0 = \frac{-10 + p}{\sqrt{11} \cdot \sqrt{20+p}}$$

Wouldn't multiplying the denominator in this case cause us to lose roots? Can someone please elaborate on this – when can we do what?

Answer 1

For $0=a/b$, the only solutions have $a=0, b\ne 0$. By multiplying to get $0=a$, you do not lose roots. You may gain roots if it could happen that $a=b=0$.

For example, $0=x/x$ has no solution, but multiplying by $x$ we get $0=x$, which does have a solution.

Answer 2

The denominator does not contribute anything towards the roots of this function. You just have to look at the numerator and make sure it falls in the domain.

Note that $\sqrt 11\times \sqrt {20+p}$ will become invalid if $p\le-20$. Hence the domain of $p$ is $(-20,\infty)$. Equating the numerator to zero, we get $p=10$. As this value falls within the domain of $p$, it is the correct and only valid root of this expression.

Answer 3

If we have a fraction $$\frac{a}{b}=0$$ then if we multiply everything by $b$ we get $a=0$ So, when a fraction is equal to $0$, then effectively the denominator becomes irrelevant.