Question 68·Medium·Equivalent Expressions

Write $u+v = \dfrac{r+s}{r-s} + \dfrac{r-s}{r+s}$. Use a common denominator. What do you get if you multiply $(r-s)(r+s)$?

After you put $u+v$ over a common denominator, you'll get $(r+s)^2 + (r-s)^2$ in the numerator. Expand both squares and combine like terms carefully.

Once you have $u+v$ as a single fraction, write $uv$ using the same denominator and then add the numerators. Be careful with the coefficients of $r^2$ and $s^2$ when you combine terms.

Choose any convenient numbers that satisfy the conditions (nonzero and $r \neq \pm s$), for example $r=5$ and $s=2$. In Desmos, type r = 5 and s = 2 on separate lines.

In Desmos, enter u = (r + s)/(r - s) and v = (r - s)/(r + s). Then enter E = u*v + u + v. Desmos will display a numeric value for $E$ based on your chosen $r$ and $s$.

For each option A–D, type its expression into Desmos, using your $r$ and $s$ values. For example, for choice A type (r^2 + 3*s^2)/(r^2 - s^2), and similarly for the others. Compare each expression's value to the value of E.

Optionally, pick a different valid pair, such as $r=4$, $s=1$, and repeat the process. The correct choice will match the value of E for both sets of $(r,s)$, confirming which expression is equivalent to $uv + u + v$.

Start by multiplying $u$ and $v$:

The $(r+s)$ in the numerator of the first fraction cancels with the $(r+s)$ in the denominator of the second fraction, and similarly for $(r-s)$, so

So the expression we need to simplify is

Now simplify $u+v$:

Use a common denominator $(r-s)(r+s)$, which equals $r^2 - s^2$ by the difference of squares identity:

Expand the squares:

Add these:

So

We now have

To add these, rewrite $1$ as a fraction with denominator $r^2 - s^2$:

So the whole expression becomes

Now that the denominators match, add the numerators:

Simplify the numerator:

So the simplified expression is

which matches answer choice B.