Question 89·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The equation

3\sqrt{m} + 2n = p

relates the positive numbers $m$ , $n$ , and $p$ . Which equation correctly expresses $m$ in terms of $n$ and $p$ ?

A radical formula asks you to get one letter alone while leaving the other letters in the answer. Undo addition and multiplication around the square root before squaring both sides. Desmos can give you a numerical checkpoint, but the algebra gives the formula that works for every allowed value. The key trap is mistaking the square root of the target for the target itself.

Hints

Hint 1
To isolate a square root means to get it alone on one side. The $2n$ is added outside the root, so what should you subtract from both sides first?
Hint 2
The $3$ multiplies the entire square root. Once the added term is gone, divide both sides by $3$ . Have you found $m$ , or only $\sqrt m$ ?
Hint 3
A square root is not usually the number under it. Once $\sqrt m$ is alone, what operation removes the root? Apply that operation to the entire other side.

Step-by-step

Isolate the root, then square
Step 1
Remove the added term
You need $m$ alone. Since $2n$ is added to the term containing $m$ , subtract $2n$ from both sides:
$3\sqrt m=p-2n.$
Step 2
Get the square root alone
The $3$ multiplies $\sqrt m$ , so divide both sides by $3$ : $\sqrt m=\frac{p-2n}{3}$ . This gives $\sqrt m$ , not $m$ . There’s one operation left to undo.
Step 3
Check the difference in Desmos
For a numerical checkpoint, choose $n=2$ and $p=13$ . These allow positive $m$ because $13>2(2)$ . Type $n=2$ , $p=13$ , and $3\sqrt{m_1}+2n\sim p\{m_1>0\}$ ; the tilde tells Desmos to find $m_1$ . It reports $m_1=9$ under PARAMETERS. Type $\frac{p-2n}{3}$ , and Desmos prints $3$ . That’s $\sqrt m$ , not $m$ . One example checks the distinction; it doesn’t prove a formula.
Step 4
Square to express m
Now that the root is alone, square both whole sides to undo it:
$\left(\sqrt m\right)^2=\left(\frac{p-2n}{3}\right)^2.$
Simplify the left side: $m=\left(\frac{p-2n}{3}\right)^2$ . Square the entire fraction, not only its denominator. This expresses $m$ in terms of $n$ and $p$ . Choice B.

Question 89·Medium·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