Question 194·Medium·Advanced Math

Use the difference of squares pattern: multiplying $(a+4)(a-4)$ makes $+4a$ and $-4a$, which cancel. The remaining terms are $a^2$ and $-4^2$. Rewrite the given equation:

Add $16$ to both sides to isolate $a^2$:

Take square roots of both sides:

The $\pm$ means either sign: a positive number and its negative have the same square. Don't choose a sign until you use the condition on $a$.

For a sample check, choose $b=9$: the earlier equation then gives $a^2=9+16=25$. Type $b=9$, then $(a_1+4)(a_1-4)\sim b\{a_1>0\}$. The subscript tells Desmos to fit $a_1$, and the restriction keeps it positive. Under PARAMETERS, Desmos reports $a_1=5$. One sample checks your work; it doesn't prove the formula for every $b$.

Because $a$ is positive, keep the positive square-root branch and reject the negative one. So $a$ expressed in terms of $b$ is

Choice C.