Question 160·Hard·Advanced Math

The point $(1,9)$ means $h(1)=9$. Put $1$ in for the input $x$:

A number to the first power stays itself, so:

The point $(2,41)$ means $h(2)=41$. Put $2$ in for $x$:

This is a sum of squares, not the product $pq$.

Use the square-of-a-sum identity. Multiplying $(p+q)(p+q)$ produces two copies of $pq$, one from $p\cdot q$ and one from $q\cdot p$:

Replace $p+q$ with $9$ and $p^2+q^2$ with $41$ to leave only the requested product unknown:

You don't need to find $p$ and $q$ separately.

Let $x_1$ stand for the product $pq$. Type $9^2\sim41+2x_1$; $\sim$ tells Desmos to find the value that makes both sides equal. Under PARAMETERS, it shows $x_1=20$, so $pq=20$. Choice B.