Practice Radical Equations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Equations with a variable under a radical sign, solved by isolating the radical, squaring both sides, and checking for extraneous solutions.

A radical 'traps' the variable inside a square root. To free it, isolate the radical on one side, then square both sides to undo the square root. But squaring can introduce fake solutions (extraneous solutions) that do not actually satisfy the original equation, so you MUST check every answer.

Showing a random 20 of 50 problems.

Example 1

hard
Solve x2โˆ’5=2\sqrt{x^2 - 5} = 2.

Example 2

challenge
Solve x+8=x+2\sqrt{x + 8} = x + 2.

Example 3

medium
Solve x+2=xโˆ’4\sqrt{x + 2} = x - 4.

Example 4

hard
Solve x+9=xโˆ’3\sqrt{x + 9} = x - 3.

Example 5

medium
Solve 3xโˆ’5=x+1\sqrt{3x - 5} = \sqrt{x + 1}.

Example 6

challenge
Solve 3x+1โˆ’1=x\sqrt{3x + 1} - 1 = \sqrt{x} for the valid root.

Example 7

medium
Solve x+2+3=7\sqrt{x + 2} + 3 = 7.

Example 8

hard
Solve 2x+33=3\sqrt[3]{2x + 3} = 3.

Example 9

medium
Solve x=โˆ’3\sqrt{x} = -3.

Example 10

hard
When is an extraneous solution possible in a radical equation?

Example 11

medium
Solve 5xโˆ’1=3x+7\sqrt{5x - 1} = \sqrt{3x + 7}.

Example 12

medium
Solve x+7=x+1\sqrt{x + 7} = x + 1.

Example 13

easy
Solve x+3=5\sqrt{x + 3} = 5.

Example 14

easy
Solve x=5\sqrt{x} = 5.

Example 15

challenge
Solve 2x+5โˆ’x+2=1\sqrt{2x + 5} - \sqrt{x + 2} = 1.

Example 16

medium
Solve x+7=x+1\sqrt{x + 7} = x + 1.

Example 17

challenge
Solve 3x+1+xโˆ’1=4\sqrt{3x + 1} + \sqrt{x - 1} = 4.

Example 18

medium
Solve x+5=2\sqrt{x} + 5 = 2.

Example 19

easy
Solve x+5=3\sqrt{x + 5} = 3.

Example 20

easy
Solve xโˆ’1=3\sqrt{x} - 1 = 3.