Question 1

Which of the following best describes a rational equation?

Accepted Answer

An equation that contains a variable in the denominator of a rational expression.

Question 2

What is the first step when solving a rational equation such as $$\frac{2x}{x-3} = 4$$?

Accepted Answer

Multiply both sides by the least common denominator (LCD).

Question 3

Solve the equation $$\sqrt{x+12} = x$$ and select the correct solution.

Accepted Answer

$$x = 6$$

Question 4

When solving equations with radicals, why is it important to check all proposed solutions in the original equation?

Accepted Answer

Because extraneous solutions may be introduced when both sides are squared.

Question 5

Solve the equation $$\sqrt{3x-1} + \sqrt{x-2} = 5$$ and select the correct solution.

Accepted Answer

$$x = 5$$

Question 6

Which of the following is the correct solution set for $$x^{2/3} = 36$?

Accepted Answer

$$x = 216$$

Question 7

Given the equation $$5x^{2/3}$$ - $$12x^{1/3} = -4$, what substitution can be used to convert it into a quadratic equation?

Accepted Answer

Let $$u = x$$^{1/3}$$

Question 8

Solve the equation $$x^{4/3}$$ - $$7x^{2/3} + 12 = 0$ and select the correct solution set.

Accepted Answer

$$x = 8, x = 27$$

Question 9

Which of the following is a necessary step when solving an equation containing two radicals, such as $$\sqrt{x+1} + \sqrt{x-2} = 5$$?

Accepted Answer

Isolate one radical before squaring both sides.

Question 10

Solve the equation $$\sqrt[3]{x+2} = x-2$$ and select the correct solution.

Accepted Answer

$$x = 6$$

Question 11

A chemical reaction rate is modeled by the equation $$\sqrt{2x+1} = x+1. $$What is the correct value of $$x$$ that satisfies this equation?

Accepted Answer

$$x = 1$$

Question 12

Which of the following is the correct solution set for $$x^4$$ - $$7x^2 + 12 = 0$?

Accepted Answer

$$x = 2, -2, 3, -3$$

Question 13

If $$x = 2 is a $$solution to a rational equation, what must you do before accepting it as a valid solution?

Accepted Answer

Check that it does not make any denominator zero in the original equation.

Question 14

Solve the equation $$\sqrt{4x+12} = x$$ and select the correct solution.

Accepted Answer

$$x = 6$$

Question 15

Which of the following equations is quadratic in form?

Accepted Answer

$$x^4$$ - $$7x^2 + 12 = 0$

Question 16

A medical dosage formula is modeled by $$\sqrt{2x+1} = x+1. $$What is the correct value of $$x$$ that satisfies this equation?

Accepted Answer

$$x = 1$$

Question 17

Which of the following is a common mistake when solving equations with radicals?

Accepted Answer

Failing to check for extraneous solutions.

Question 18

Solve the equation $$x^{2/3}$$ - $$7x^{1/3} + 12 = 0$ and select the correct solution set.

Accepted Answer

$$x = 8, x = 27$$

Question 19

Which of the following is the correct solution for $$x^{1/2} = 9$?

Accepted Answer

$$x = 81$$

Equations and Applications: Rational, Radical, and Quadratic Forms

Study Guide - Practice Questions