Extraneous solutions of radical equations

Practice some problems before going into the exercise.

Introduction

In this article we will practice some problems that involve thinking about the conditions for obtaining extraneous solutions while solving radical equations.

Practice question 1

Caleb is solving the following equation for

x

x = \sqrt{x + 2} + 7

His first few steps are given below.

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

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

Practice question 2

Elena is solving the following equation for

x

\sqrt[3]{3 x - 5} + 2 = 7

Her first few steps are given below.

\begin{aligned} \sqrt[3]{3 x - 5} & = 5 \\ {(\sqrt[3]{3 x - 5})}^{3} & = (5)^{3} \\ 3 x - 5 & = 125 \end{aligned}

Is it necessary for Elena to check her answers for extraneous solutions?

Practice question 3

Addison solves the equation below by first squaring both sides of the equation.

2 x - 1 = \sqrt{8 - x}

What extraneous solution does Addison obtain?

x =

Practice question 4

Which value for the constant $d$ ‍ makes $x = - 1$ ‍ an extraneous solution in the following equation?

\sqrt{8 - x} = 2 x + d

d =

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