Radical Equations Worksheet- Practice Problems and Solutions

What Radical Equations Actually Are

A radical equation is any equation where the variable sits under a square root (or other root). The catch: you can't just solve them like regular equations. The square root complicates everything.

The standard form looks like this: √(x + 3) = 7

Most students see these problems and freeze. That's not a confidence problem—that's a process problem. These equations follow a specific set of steps, and skipping any of them gives you wrong answers.

The Core Rule You Can't Ignore

When you square both sides of an equation, you don't create equivalent equations. You create implications.

What does that mean in practice? Squaring can introduce extraneous solutions—answers that look correct but actually don't work when you plug them back in.

This is why every radical equation problem requires one non-negotiable step: check your solutions in the original equation.

How to Solve Radical Equations

Here's the process that actually works:

Step 1: Isolate the Radical

Move the square root term so it stands alone on one side. If you have 2 + √(x-1) = 10, subtract 2 first to get √(x-1) = 8.

Step 2: Square Both Sides

Eliminate the radical by squaring everything. (√(x-1))² = 8² becomes x - 1 = 64.

Step 3: Solve the Remaining Equation

Now it's basic algebra. Add 1 to both sides: x = 65.

Step 4: Check Your Answer

Plug x = 65 back into the original equation: 2 + √(65-1) = 2 + √64 = 2 + 8 = 10. The left side equals 10, so this solution works.

Common Mistakes That Ruin Your Answers

• Forgetting to isolate first: You must get the radical by itself before squaring. Squaring 2 + √x doesn't give you (√x)².
• Only squaring one term: When you square both sides, everything gets squared. (x + 3)² is x² + 6x + 9, not just x² + 3.
• Skipping the check: This is how extraneous solutions slip through. Always verify.
• Missing domain restrictions: The expression under an even root must be ≥ 0. If your solution makes the radicand negative, it's automatically wrong.

Practice Problems with Solutions

Problem 1

Solve: √(2x + 5) = 9

Solution:

Square both sides: 2x + 5 = 81

Subtract 5: 2x = 76

Divide by 2: x = 38

Check: √(2(38) + 5) = √(76 + 5) = √81 = 9 ✓

Problem 2

Solve: √(x + 4) + 6 = 10

Solution:

Isolate the radical: √(x + 4) = 4

Square both sides: x + 4 = 16

Solve: x = 12

Check: √(12 + 4) + 6 = √16 + 6 = 4 + 6 = 10 ✓

Problem 3

Solve: √(3x - 2) = x - 2

Solution:

Square both sides: 3x - 2 = (x - 2)²

Expand: 3x - 2 = x² - 4x + 4

Rearrange: 0 = x² - 7x + 6

Factor: 0 = (x - 6)(x - 1)

Solutions: x = 6 or x = 1

Check x = 6: √(3(6) - 2) = √16 = 4, and 6 - 2 = 4 ✓

Check x = 1: √(3(1) - 2) = √1 = 1, and 1 - 2 = -1 ✗

Only x = 6 works. The x = 1 solution is extraneous.

Problem 4

Solve: √(x + 7) = √(2x - 3)

Solution:

Square both sides: x + 7 = 2x - 3

Solve: 7 + 3 = 2x - x

x = 10

Check: √(10 + 7) = √17 and √(20 - 3) = √17 ✓

When You Have Multiple Radicals

Some equations have radicals on both sides. The process stays the same, but you might need to square twice:

√(x + 5) + √(x - 1) = 6

First, isolate one radical: √(x + 5) = 6 - √(x - 1)

Square both sides, simplify, then square again. It's tedious. That's why worksheets exist—to build your speed without losing accuracy.

Solving Methods Comparison

Getting Started with Your Worksheet

Don't try to memorize everything at once. Here's what actually works:

• Start with simple problems that have one radical and no extra terms. Get those right before adding complexity.
• Write every step. Don't try to do it in your head. The written process is how you catch mistakes.
• Check every answer. Not after the worksheet—after every single problem. This habit alone will save you from lost points on tests.
• Time yourself once you're confident. The goal is accuracy first, speed second.

What to Do When You're Stuck

If a problem isn't working, check these in order:

1. Did you isolate the radical completely before squaring?
2. Did you square every term on both sides?
3. Did you combine like terms correctly after expanding?
4. Does your solution satisfy the domain restrictions?
5. Did you plug it back into the original equation?

One of those five questions catches 95% of all errors.

Why Worksheets Actually Help

You can read about solving radical equations for hours. That doesn't mean you can solve them. Math isn't a reading comprehension subject.

You need reps. You need to see problems you got wrong, figure out which step broke down, and try again. A worksheet gives you that structure when you don't have a teacher watching over your shoulder.

Focus on problems 1-20 on any standard worksheet. If you can get 18 of those right without help, you're ready for harder problems. If not, you know exactly where to drill.