An equation with the unknown inside square roots is a radical equation. First find the allowed inputs, since the number under a square root can't be negative. Then use a Desmos regression, which finds an unknown number that makes the equation true. Keep the original equation intact: squaring a sum term by term leaves out a middle term and can lead to a wrong answer.
Hints
- Hint 1
A square root needs a nonnegative number inside it. Check the expressions under both roots. Which one puts the stronger limit on ?
- Hint 2
As an allowed value of increases, both square roots increase. Their sum can hit the given total at most once, so one solution will settle the question.
- Hint 3
To solve with a Desmos regression, replace with and with . This tells Desmos to find a number that makes the original two sides equal.
Step-by-step
Solve the original equation in Desmos
Step 1Find the allowed inputs
The number inside a square root can't be negative. The tighter condition comes from : , so . Those inputs also make nonnegative.
- Step 2
See why one result is enough
For , increasing increases both numbers under the roots, so both roots increase. Their sum only rises, so it can equal at most once.
- Step 3
Solve without changing the equation
Type . The tells Desmos to find a number rather than graph using ; tells it to fit the two sides. Under PARAMETERS, Desmos reports . Because this fits the original equation and there can be at most one solution, the value of is . Choice C.