Question 187·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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If $\sqrt{x+4}+\sqrt{x-1}=5$ , what is the value of $x$ ?

In a radical equation, the unknown sits under a root. First check which inputs make both roots real, then use a one-variable regression in Desmos to solve the original equation. Keeping the original equation intact avoids false answers that squaring can create. A nearby whole number isn't enough; it must make the two sides equal.

Hints

Hint 1
A square root is real only when the number under it is at least $0$ . Start with $x-1$ : which values of $x$ keep it nonnegative? Do those values also work inside the other root?
Hint 2
A Desmos regression finds the input that makes the two sides match. Replace $x$ with $x_1$ and $=$ with $\sim$ . Desmos reports the value it finds under PARAMETERS.

Step-by-step

Solve the original equation in Desmos
Step 1
Find the allowed inputs
A square root needs a nonnegative number inside it. Require the expression under the second root to be at least $0$ :
$x-1\ge0.$
Add $1$ to both sides:
$x\ge1.$
These values also keep $x+4$ nonnegative.
Step 2
Find the value that makes both sides equal
Type $\sqrt{x_1+4}+\sqrt{x_1-1}\sim5$ . The subscripted $x_1$ tells Desmos to fit a value, and $\sim$ tells it to match the sides. Under PARAMETERS, Desmos shows $x_1=5$ . Check it in the original equation: $\sqrt{5+4}+\sqrt{5-1}=3+2=5$ . Both roots grow as $x$ grows, so there isn't another match. The value of $x$ is $5$ . Choice C.

Question 187·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this