Question 65·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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For positive numbers $a$ and $b$ , the equation $(a+4)(a-4)=b$ holds. Which equation correctly expresses $a$ in terms of $b$ ?

The paired factors $(a+4)(a-4)$ signal a difference of squares: their middle terms cancel. Rewrite the product, isolate $a^2$ , and take square roots. A Desmos check with one value of $b$ can help you catch a slip, but the algebra gives a formula for every $b$ . Use the positive condition to choose the right root.

Hints

Hint 1
A difference of squares has the pattern $(u+v)(u-v)=u^2-v^2$ . The terms containing both $u$ and $v$ cancel. What remains when you multiply the paired factors containing $a$ and $4$ ?
Hint 2
Once the product is rewritten, get $a^2$ alone by doing the same operation to both sides of the equation. What must you add to undo the subtraction of $16$ ?
Hint 3
A square root reverses squaring, but $5$ and $-5$ both square to $25$ . Use the condition that $a$ is positive to decide which sign belongs in your formula.

Step-by-step

Rewrite the product and solve for a
Step 1
Recognize the difference of squares
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:
$a^2-16=b.$
Step 2
Get the square alone
Add $16$ to both sides to isolate $a^2$ :
$a^2=b+16.$
Step 3
Undo the square
Take square roots of both sides:
$a=\pm\sqrt{b+16}.$
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 $a$ .
Step 4
Check a positive solution in Desmos
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 $b$ .
Step 5
Choose the root that fits
Because $a$ is positive, keep the positive square-root branch and reject the negative one. So $a$ expressed in terms of $b$ is
$a=\sqrt{b+16}.$
Choice C.

Question 65·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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