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$Caleb is solving the following equation for x. x=sqrt(x+2)+7 His first few steps are given below. {:[x-7=sqrt(x+2)],[(x-7)^(2)=(sqrt(x+2))^(2)],[x^(2)-14 x+49=x+2]:} Is it necessary for Caleb to check his answers for extraneous solutions? Choose 1 answer: (A) Yes (B) No$

Caleb is solving the following equation for $x$ . $\newline$ $x=\sqrt{x+2}+7$ $\newline$ His first few steps are given below. $\newline$ $\begin{aligned} x-7 & =\sqrt{x+2} \\ (x-7)^{2} & =(\sqrt{x+2})^{2} \\ x^{2}-14 x+49 & =x+2 \end{aligned}$ $\newline$ Is it necessary for Caleb to check his answers for extraneous solutions? $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No

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Solve a system of equations using any method: word problems

Full solution

Q. Caleb is solving the following equation for

x

.

\newline

x=\sqrt{x+2}+7

\newline

His first few steps are given below.

\newline

\begin{aligned} x-7 & =\sqrt{x+2} \\ (x-7)^{2} & =(\sqrt{x+2})^{2} \\ x^{2}-14 x+49 & =x+2 \end{aligned}

\newline

Is it necessary for Caleb to check his answers for extraneous solutions?

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Isolating the square root: Caleb starts with the equation $x = \sqrt{x + 2} + 7$ and isolates the square root on one side by subtracting $7$ from both sides. $\newline$ $x - 7 = \sqrt{x + 2}$
Eliminating the square root: Next, Caleb squares both sides of the equation to eliminate the square root. $(x - 7)^2 = (\sqrt{x + 2})^2$
Expanding and simplifying: Caleb then expands the left side of the equation and simplifies the right side. $x^2 - 14x + 49 = x + 2$
Bringing all terms together: Caleb needs to bring all terms to one side to set the equation to zero and solve for $x$ . $x^2 - 14x + 49 - x - 2 = 0$ $x^2 - 15x + 47 = 0$
Checking for extraneous solutions: Caleb would then solve the quadratic equation for $x$ . However, before doing that, we need to address the question of whether it is necessary to check for extraneous solutions. Since the original equation involves a square root and we squared both sides of the equation, there is a possibility that squaring both sides could introduce extraneous solutions. Therefore, it is necessary for Caleb to check his answers for extraneous solutions.

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