Question 178·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation $\left(\dfrac{p}{3}\right)^{2} = r$ relates the positive numbers $p$ and $r$ . Which equation correctly expresses $p$ in terms of $r$ ?

To rearrange a nonlinear formula like this one, undo the square and then undo the division around the letter you want. The positive condition tells you which square-root sign to keep. A Desmos check at one positive value can help catch a sign mistake, but the algebra gives a rule that works for every allowed value.

Hints

Hint 1
A square root undoes a square, but a positive number and its negative have the same square. Since $p$ is positive, which sign can $p/3$ have?
Hint 2
After you undo the square, $p$ is still divided by $3$ . Use the balance rule: multiply both sides by $3$ to get $p$ alone.

Step-by-step

Undo the square, then the division
Step 1
Check the positive solution at one value
Pick $r=4$ , a positive perfect square, to check the sign. Type $r=4$ and $\left(\frac{p_{1}}{3}\right)^2\sim r\left\{p_{1}>0\right\}$ . The $\sim$ asks Desmos to find $p_1$ , and the braces require it to be positive. Under PARAMETERS, Desmos shows $p_1=6$ . That's a check for one value of $r$ , not yet a formula.
Step 2
Undo the square
Since $p$ is positive, $p/3$ is positive. Taking a square root undoes the square. Squaring hides the sign, but a negative root would make $p/3$ negative, so keep the positive one:
$\frac{p}{3}=\sqrt r.$
Use the sign condition to choose the square-root branch.
Step 3
Get p alone
Multiply both sides by $3$ to undo the division:
$p=3\sqrt r.$
The $3$ goes outside the root because it multiplies the whole value of $\sqrt r$ . This expresses $p$ in terms of $r$ . Choice D.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 178·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this