Radical Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radical Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Free the variable from under the root by isolating and squaring, then reject any extraneous solution.

Common stuck point: The procedure for radical equations is the easy part; the trap is skipping the check. Asking "Is the unknown trapped under a radical that I undo by squaring, and did I check for extraneous roots?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the unknown trapped under a radical that I undo by squaring, and did I check for extraneous roots?

Worked Examples

Example 1

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

Answer

x=22x = 22

First step

1
Step 1: Square both sides: x+3=25x + 3 = 25.

Full solution

  1. 2
    Step 2: Subtract 3: x=22x = 22.
  2. 3
    Step 3: Check: 22+3=25=5\sqrt{22 + 3} = \sqrt{25} = 5 ✓
To solve a radical equation, isolate the radical then square both sides. Always check the solution because squaring can introduce extraneous solutions.

Example 2

hard
Solve 2x+1=x−1\sqrt{2x + 1} = x - 1.

Example 3

medium
Solve x+6=x\sqrt{x + 6} = x.

Example 4

hard
Solve x+1+x−4=5\sqrt{x + 1} + \sqrt{x - 4} = 5.

Example 5

hard
Solve x+4−x−3=1\sqrt{x + 4} - \sqrt{x - 3} = 1.

Example 6

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve x=7\sqrt{x} = 7.

Example 2

medium
Solve 3x−2=4\sqrt{3x - 2} = 4.

Example 3

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

Example 4

easy
Solve x=7\sqrt{x} = 7.

Example 5

easy
Solve x+1=4\sqrt{x + 1} = 4.

Example 6

easy
Solve 2x=6\sqrt{2x} = 6.

Example 7

easy
Solve x−3=2\sqrt{x - 3} = 2.

Example 8

easy
Solve 3x+1=5\sqrt{3x + 1} = 5.

Example 9

easy
Solve x+2=6\sqrt{x} + 2 = 6.

Example 10

easy
Solve x−1=3\sqrt{x} - 1 = 3.

Example 11

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

Example 12

medium
Solve x+6=x\sqrt{x + 6} = x.

Example 13

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

Example 14

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

Example 15

medium
Solve x−1+3=x\sqrt{x - 1} + 3 = x.

Example 16

medium
Solve 4x+5=2x−1\sqrt{4x + 5} = 2x - 1 for the valid root.

Example 17

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

Example 18

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

Example 19

medium
Solve 2x=x−32\sqrt{x} = x - 3 for the valid root.

Example 20

challenge
Solve x+5−x=1\sqrt{x + 5} - \sqrt{x} = 1.

Example 21

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

Example 22

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

Example 23

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

Example 24

easy
Solve 4x=6\sqrt{4x} = 6.

Example 25

easy
Solve x−4=3\sqrt{x - 4} = 3.

Example 26

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

Example 27

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

Example 28

medium
Solve 2x=102\sqrt{x} = 10.

Example 29

medium
Solve 5x−1=7\sqrt{5x - 1} = 7.

Example 30

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

Example 31

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

Example 32

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

Example 33

hard
Solve x+5−x=1\sqrt{x + 5} - \sqrt{x} = 1.

Example 34

hard
Solve 2x+7=x−2\sqrt{2x + 7} = x - 2.

Example 35

hard
Solve x−13=2\sqrt[3]{x - 1} = 2.

Example 36

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

Example 37

medium
Solve x=−3\sqrt{x} = -3.

Example 38

medium
Solve 4x+1−5=0\sqrt{4x + 1} - 5 = 0.

Example 39

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

Example 40

hard
Solve x2−5=2\sqrt{x^2 - 5} = 2.

Example 41

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

These ideas may be useful before you work through the harder examples.

radical operationssolving linear equations