Question 1

Which of the following best describes a radical equation?

Accepted Answer

An equation where the variable is inside a root, such as a square root or cube root.

Question 2

What is the first step when solving a radical equation such as $$\sqrt{x+3} = 5$$?

Accepted Answer

Isolate the radical on one side of the equation.

Question 3

Solve the equation $$\sqrt{2x - 1} = 3. $$What is the solution?

Accepted Answer

$$x = 4$$

Question 4

After solving a radical equation by raising both sides to an even power, why must you check all proposed solutions in the original equation?

Accepted Answer

To verify that no extraneous solutions were introduced.

Question 5

Solve the equation $$\sqrt{x + 4} = x. $$Which of the following is a valid solution?

Accepted Answer

Question 6

Which of the following is an extraneous solution for the equation $$\sqrt{x} = -3$$?

Accepted Answer

There is no real solution.

Question 7

Solve the equation $$\sqrt{3x - 2} = x - 2. $$After checking for extraneous solutions, what is the valid solution?

Accepted Answer

$$x = 4$$

Question 8

A medical researcher models the concentration of a drug in the bloodstream with $$C = \sqrt{t + 5}$$, where $$C is $$the concentration and $$t is $$time in hours. If the concentration reaches 7 units, how many hours have passed?

Accepted Answer

$$t = 44$$

Question 9

Solve the equation $$\sqrt{2x + 5} = x. $$After squaring both sides and checking for extraneous solutions, which value(s) of $$x$$ are valid?

Accepted Answer

$$x = 1$$

Question 10

Which step is necessary when solving $$\sqrt{x - 1} + 2 = x$$?

Accepted Answer

Isolate $$\sqrt{x - 1} by $$subtracting 2 from both sides.

Question 11

Solve the equation $$\sqrt{x + 6} = x - 2. $$Which of the following is a valid solution after checking for extraneous roots?

Accepted Answer

$$x = 4$$

Question 12

A chemical reaction is modeled by $$\sqrt{y} = 5$$, where $$y is $$the concentration of a reactant. What is the concentration?

Accepted Answer

$$y = 25$$

Question 13

If $$\sqrt{2x - 3} = x$$, what is the solution after checking for extraneous roots?

Accepted Answer

$$x = 3$$

Question 14

Which of the following statements about extraneous solutions is correct?

Accepted Answer

Extraneous solutions occur when raising both sides of a radical equation to an even power.

Question 15

Solve the equation $$\sqrt{x} + 2 = 6. $$What is the value of $$x$$?

Accepted Answer

$$x = 16$$

Question 16

A student solves $$\sqrt{x + 1} = -4$$ and finds $$x = 15. $$What is the error in their solution?

Accepted Answer

The square root of any real number cannot be negative.

Question 17

Solve the equation $$\sqrt{4x - 7} = 5. $$What is the value of $$x$$?

Accepted Answer

$$x = 8$$

Question 18

Which of the following is the correct process for solving $$\sqrt{x + 2} = x - 2$$?

Accepted Answer

Isolate the radical, square both sides, solve the resulting equation, and check for extraneous solutions.

Question 19

A real-world scenario: The time $$t (in $$hours) it takes for a bacteria population to reach a certain size is modeled by $$\sqrt{t} = 6. $$What is the value of $$t$$?

Accepted Answer

$$t = 36$$

Solving Radical Equations in College Algebra

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