Question 204·Medium·Advanced Math

Type $r=2$ and $s=1$. Both are nonzero, and $2$ is neither $1$ nor $-1$, so neither given denominator is zero. Their squares are different, which helps distinguish the choices.

Type the given definitions, then $uv+u+v$. Desmos shows $u=3$, $v=0.3333$, and $uv+u+v=4.3333$. Keep those same $r$ and $s$ values when you check the choices.

Type each choice below the original. In order, Desmos gives $2.3333$, $4.3333$, $0.2$, and $1.8$. Three choices disagree at an allowed input, so none of those can be equivalent. The remaining expression is $\frac{3r^2+s^2}{r^2-s^2}$. Choice B.

$u$ and $v$ are reciprocals: each fraction flips the other's top and bottom. Multiply them, then cancel the nonzero factors:

A common denominator is the same bottom for both fractions. Multiply each top and bottom by the factor its bottom is missing, then add the tops: $u+v=\frac{(r+s)^2+(r-s)^2}{(r-s)(r+s)}$. Don't add the original denominators to add fractions.

A square multiplies the whole parentheses by itself, so each square has a middle term. Expand both:

The $+2rs$ and $-2rs$ terms cancel, while both $r^2$ terms and both $s^2$ terms add. Combine them:

The denominator is a difference of squares: $(r-s)(r+s)=r^2-s^2$, because the $+rs$ and $-rs$ terms cancel. Rewrite it: $u+v=\frac{2r^2+2s^2}{r^2-s^2}$. The minus sign belongs in the denominator.

Since $uv=1$, replace the product in the expression the question asks about: $uv+u+v=1+\frac{2r^2+2s^2}{r^2-s^2}$. Keep the $1$; finding $u+v$ alone stops too early.

Write $1$ as a fraction with denominator $r^2-s^2$: $uv+u+v=\frac{r^2-s^2}{r^2-s^2}+\frac{2r^2+2s^2}{r^2-s^2}$. The denominators now match.

The fractions have the same denominator, so add their tops:

Combine the $r^2$ terms and the $s^2$ terms: $uv+u+v=\frac{3r^2+s^2}{r^2-s^2}$. This is the equivalent expression. Choice B.