Question 1

Solve for $x$: $\sqrt{x} + 2 = x$. Which of the following is a real solution?

Accepted Answer

$x = 4$

Question 2

Which of the following statements is TRUE about solving equations involving square roots?

Accepted Answer

Squaring both sides can introduce extraneous solutions that must be checked in the original equation.

Question 3

Solve for all real solutions: $x^{2/3} = 1/2$. What is the correct solution?

Accepted Answer

$x = \left(\dfrac{1}{2}\right)^{3/2}$

Question 4

Which of the following is the correct first step to solve the equation $\sqrt{3x-2} - \sqrt{x} = 2$?

Accepted Answer

Isolate one of the square roots on one side of the equation.

Question 5

Solve for $x$: $x^{-1/2} = \dfrac{1}{2}$. What is the solution?

Accepted Answer

$x = 4$

Question 6

Which of the following equations is of quadratic type and can be solved by substitution?

Accepted Answer

$x^4 + 14x^2 - 32 = 0$

Question 7

Solve for $x$: $|3x + 4| = 12$. Which of the following is NOT a solution?

Accepted Answer

$x = 4$

Question 8

Given $x^{4/3} = 625$, what are the real solutions for $x$?

Accepted Answer

$x = \pm 25$

Question 9

Which of the following best describes the process for solving $|x| = |x-1|$?

Accepted Answer

Set $x = x-1$ and $x = -(x-1)$, then solve both equations.

Question 10

Solve for $x$: $x^4 - 81 = 0$. Which of the following are all real solutions?

Accepted Answer

$x = \pm 3, \pm 9$

Question 11

A medical test result is modeled by the equation $|x-4| - 1 = -4x$. What is the first step to solve for $x$?

Accepted Answer

Isolate the absolute value expression on one side.

Question 12

Solve for $x$: $(x-2)^{-1/2} = \dfrac{1}{\sqrt{3}}$. What is the solution?

Accepted Answer

$x = 5$

Question 13

Which of the following is a necessary step when solving $\sqrt{x-1} = x-7$?

Accepted Answer

Isolate the square root and then square both sides, checking for extraneous solutions.

Question 14

Solve for $x$: $x^2 - 4\sqrt{x} + 3 = 0$. Which substitution is most helpful?

Accepted Answer

Let $y = \sqrt{x}$, then solve the resulting quadratic in $y$.

Question 15

A chemical reaction is modeled by $|5x-3| = |2x+7|$. What are the possible equations to solve for $x$?

Accepted Answer

$5x-3 = 2x+7$ and $5x-3 = -(2x+7)$

Question 16

Solve for $x$: $(x-2)^{-5/2} = 32$. What is the solution?

Accepted Answer

$x = 3$

Question 17

Which of the following is the correct interpretation of $x^{m/n} = k$?

Accepted Answer

$x^{m/n}$ means the $n$th root of $x$ raised to the $m$th power.

Question 18

Solve for $x$: $|x^2 - 6| = 5x$. Which of the following is a correct approach?

Accepted Answer

Set $x^2 - 6 = 5x$ and $x^2 - 6 = -5x$, then solve both equations.

Question 19

Which of the following is a possible extraneous solution when solving $\sqrt{x} = -3$?

Accepted Answer

$x = 9$

Question 20

Solve for $x$: $x^{2/3} = 4$. What are the real solutions?

Accepted Answer

$x = 8$ and $x = -8$

Solving Miscellaneous Equations: Square Roots, Rational Exponents, Quadratic Type, and Absolute Values

Study Guide - Practice Questions