Debug

How to Solve Radical Equations

Video Tutorial and practice problems

How To solve Radical Equations

• 1) Isolate radical on one side of the equation
• 2) Square both sides of the equation to eliminate radical
• 3) Simplify and solve as you would any equations
• 4) Substitute answers back into original equation to make sure that your solutions are valid (there could be some extraneous roots that do not satisfy the original equation and that you must throw out)

The video below and our examples explain these steps and you can then try our practice problems below.

Practice Problems

Problem 1
Step 1

Step 2

Square both sides.

Step 3

Solve expression.

x = 10
Step 4

Substitute answer into original radical equation to verify that the answer is a real number.

Problem 2
Step 1

Step 2

Square both sides.

Step 3

Solve expression.

3x = 23
Step 4

Substitute answer into original radical equation to verify that the answer is a real number.

Problem 3
Step 1

Step 2

Square both sides.

Step 3

Solve expression.

0 =(x - 4)(x - 5)
x = 4, x = 5
Step 4

Substitute answer into original radical equation to verify that the answer is a real number.

$$\sqrt{3x -11} = 3x -x \\ \sqrt{3 (\color{Red}{4}) -11} = 3 \cdot (\color{Red}{4}) -\color{Red}{4} \\ \sqrt{1} = 8 \\ 1 \color{red}{ \ne } 8$$

Therefore, reject 4 as a solution, check 5.

$$\sqrt{3x -11} = 3x -x \\ \sqrt{3 (\color{Red}{5}) -11} = 3(\color{Red}{5}) - \color{Red}{5} \\ \sqrt{15 -11} = 15 - 5 \\ \sqrt{15 -11} = 15 - 5 \\ \sqrt{4} = 10 \\ 2 = 10 \\ \color{red}{ \ne } 10$$

Therefore, reject 5 as a solution.

Since both our solutions were rejected, there are no real solutions to this equation.