Solving Radical Equations

Learn how to solve equations containing square roots by isolating the radical and squaring both sides.

Advanced25 minLesson
Printable worksheet

Definition

A radical equation is an equation in which the variable appears under a radical sign (usually a square root).
Examples of radical equations:
x=5\sqrt{x} = 5
x+3=7\sqrt{x + 3} = 7
2x−1=x−2\sqrt{2x - 1} = x - 2
The solving process:
  1. 1.Isolate the radical on one side
  2. 2.Square both sides to eliminate the radical
  3. 3.Solve the resulting equation
  4. 4.Check all solutions (critical step!)
Why check? Squaring can introduce extraneous solutions - values that satisfy the squared equation but NOT the original.

Try it now

Solve: x=4\sqrt{x} = 4

Worked Examples

Solve: x=6\sqrt{x} = 6
1

The radical is already isolated

x=6\sqrt{x} = 6 → Ready to square
2

Square both sides

(x)2=62(\sqrt{x})^2 = 6^2 → x=36x = 36
3

Check the solution

36=6\sqrt{36} = 6 ✓ → Valid solution

Common Mistakes

Forgetting to check for extraneous solutions

Why it's wrong: Squaring both sides can introduce false solutions. For example, in x=−3\sqrt{x} = -3, squaring gives x=9x = 9, but 9=3≠−3\sqrt{9} = 3 \neq -3.

Correct: Always substitute your answer back into the ORIGINAL equation to verify.

Squaring before isolating the radical

Why it's wrong: If the radical is not alone, squaring creates a more complex equation. For x+2=5\sqrt{x} + 2 = 5, squaring gives x+4x+4=25x + 4\sqrt{x} + 4 = 25, which is harder to solve.

Correct: First isolate: x=3\sqrt{x} = 3, then square: x=9x = 9.

Squaring each term instead of the whole side

Why it's wrong: A side that is a sum or difference must be squared as a whole. In x+3=x−3\sqrt{x + 3} = x - 3, the right side does NOT become x2−9x^2 - 9 or x2+9x^2 + 9.

Correct: Square the whole side: (x−3)2=x2−6x+9(x - 3)^2 = x^2 - 6x + 9

Assuming square root can be negative

Why it's wrong: The principal square root is always non-negative. 25=5\sqrt{25} = 5, not ±5\pm 5.

Correct: If x=−k\sqrt{x} = -k where k>0k > 0, there is no solution.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Solve Radical Equations | Step-by-Step Examples

by Understand The Math

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Interactive Visual

Function graph

-4-20246810-4-20246xy(1, -2)(6, 3)

Tap or hover over the graph to read off values.

y = √(x + 3)y = x - 3y = -√(x + 3)
  • Intersection point: (1, -2)
  • Intersection point: (6, 3)

Equation Solver

Solve for xx:

x+5=4\sqrt{x + 5} = 4
1
x+5=4\sqrt{x + 5} = 4
Square both sidesSquaring removes the square root

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

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Practice Problems

16 problems
Problem 1 of 16
Easy
Solve: x=4\sqrt{x} = 4

Why It Matters

Radical equations appear throughout science and engineering:
  • Physics: The period of a pendulum is T=2πLgT = 2\pi\sqrt{\frac{L}{g}}, where LL is the length
  • Distance formula: Finding coordinates when (x2−x1)2+(y2−y1)2=d\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = d
  • Engineering: Stress calculations, wave equations, and electrical circuits
  • Finance: Time value of money formulas involving roots
Understanding how to solve these equations opens doors to physics, engineering, and advanced mathematics!

Real World Applications

Pendulum Period

The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$ where $L$ is the length and $g = 9.8$ m/s² is gravity.

Example:

If a clock pendulum needs a period of 2 seconds, what length should it be? Solve 2=2πL9.82 = 2\pi\sqrt{\frac{L}{9.8}}

1Try It Yourself

A grandfather clock needs a pendulum with a period of exactly 1 second.

What length pendulum is needed, in meters, rounded to 2 decimal places? Use T=2πL9.8T = 2\pi\sqrt{\frac{L}{9.8}}

Step 1: Write the mathematical expression

Start with 1=2πL9.81 = 2\pi\sqrt{\frac{L}{9.8}}

Free Fall Distance

The distance an object falls is related to time by $d = \frac{1}{2}gt^2$, or equivalently $t = \sqrt{\frac{2d}{g}}$.

Example:

How long does it take for a ball to fall 45 meters? Solve t=2×459.8t = \sqrt{\frac{2 \times 45}{9.8}}

2Try It Yourself

A rock is dropped from a cliff and takes 3 seconds to hit the ground.

How high is the cliff, in meters? Use t=2d9.8t = \sqrt{\frac{2d}{9.8}}

Step 1: Write the mathematical expression

Solve 3=2d9.83 = \sqrt{\frac{2d}{9.8}} for dd

Key Takeaways

  • 1A radical equation contains a variable under a radical sign (like x\sqrt{x})
  • 2To solve: isolate the radical, square both sides, solve, then CHECK
  • 3Squaring can create extraneous solutions that don't work in the original equation
  • 4A square root equals a negative number has no real solution
  • 5Always verify solutions by substituting back into the original equation

Frequently Asked Questions

Squaring is not a reversible operation - both 323^2 and (−3)2(-3)^2 equal 9. When we square both sides of an equation, we potentially introduce solutions from the negative case that don't satisfy the original.
Squaring is not a reversible operation - both 323^2 and (−3)2(-3)^2 equal 9. When we square both sides of an equation, we potentially introduce solutions from the negative case that don't satisfy the original.
Isolate one radical, square both sides, then isolate the remaining radical and square again. You may need to square twice. Always check your solutions!
Yes! If the equation requires a square root to equal a negative number, there is no real solution. Also, all potential solutions might be extraneous.

Glossary

Radical equation
An equation where the variable appears under a radical sign
Extraneous solution
A value that satisfies the transformed equation but not the original equation
Principal square root
The non-negative square root of a number, denoted by \sqrt{}
Isolate
To get a term alone on one side of an equation

Formula Card

Basic Method

If f(x)=g(x), then f(x)=[g(x)]2\text{If } \sqrt{f(x)} = g(x), \text{ then } f(x) = [g(x)]^2

Square both sides to eliminate the radical

Check Condition

g(x)≥0 (right side must be non-negative)g(x) \geq 0 \text{ (right side must be non-negative)}

The right side must be non-negative

Domain Restriction

f(x)≥0 (expression under radical must be non-negative)f(x) \geq 0 \text{ (expression under radical must be non-negative)}

Expression under the radical must be non-negative

Today's challengeOne problem, three tries

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