Which of the following describes the solutions to the... | SAT Nonlinear equations in one variable and systems of equations in two variables Question, Answer Explained

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Question bank / Nonlinear equations in one variable and systems of equations in two variables

Math · Advanced MathHard

Which of the following describes the solutions to the equation $\sqrt{x + 7} = x - 5$ ?

$2$ only

$9$ only

$2$ and $9$

There are no solutions.

Show answer and explanation

Answer: B

Square both sides: $x + 7 = x^2 - 10x + 25$ , so $x^2 - 11x + 18 = 0$ and $(x - 2)(x - 9) = 0$ . Check each value in the original equation. For $x = 9$ : $\sqrt{16} = 4$ and $9 - 5 = 4$ , so it works. For $x = 2$ : $\sqrt{9} = 3$ but $2 - 5 = -3$ , so $2$ is extraneous. The only solution is $9$ .

Choice C includes $x = 2$ , which solves the squared equation but not the original one, since a square root can't equal $-3$ .

Why the other choices are wrong

A. x = 2 gives √9 = 3 on the left and -3 on the right, so it doesn't work. The value that works is 9.
C. Squaring both sides added the extraneous value 2. In the original equation it gives 3 = -3.
D. x = 9 works: √16 = 4 and 9 - 5 = 4.

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