Radical equations contain variables inside square roots or other roots, so ordinary algebra steps are not always enough. This guide explains how to solve equations with square roots by isolating the radical, squaring both sides, solving the resulting equation, and checking every candidate in the original equation. You will also learn how restrictions prevent invalid work, why extraneous solutions appear, and how to practice the method with increasing difficulty.
Overview
A radical equation is an equation in which a variable appears under a radical sign. For example:
√(x + 5) = 4
The most common radical equations use square roots, but the same general idea applies to cube roots and other indexed radicals. The key difference is that squaring both sides can produce a value that solves the transformed equation but not the original one. That value is called an extraneous solution.
Use this reliable workflow for square-root equations:
- Identify the expressions under every even root.
- State any restrictions suggested by the original equation.
- Isolate one radical on one side of the equation.
- Square both sides carefully.
- Solve the resulting equation, which may be linear or quadratic.
- Substitute each candidate into the original equation.
- Keep only the candidates that make the original equation true.
Do not skip the final check. Squaring is a reversible step when signs are already known, but it is not reversible for arbitrary expressions. For instance, both 3 and −3 become 9 when squared. The original equation may accept only one of them.
| Situation | Recommended method | Main caution |
|---|---|---|
| One isolated square root | Square both sides, then solve | Check the result in the original equation |
| A square root plus other terms | Move nonradical terms first | Do not square before isolating the radical |
| Two square roots | Isolate one radical, square, isolate the remaining radical, then square again | Expect possible extraneous candidates |
| A quadratic after squaring | Factor or use the quadratic formula | Every quadratic root still needs an original-equation check |
How to estimate
Before doing exact algebra, you can often estimate where a solution might be. This is useful for catching calculator or copying errors. If √(x + 5) = 4, the radicand must be 16, so x should be near 11. Estimation does not replace the solution process, but it gives you a reasonableness check.
Step 1: Find the domain restriction
For a real square root, the radicand must be greater than or equal to zero. Thus, for √(2x − 6):
2x − 6 ≥ 0, so x ≥ 3.
This restriction describes values for which the original expression is real. It is not always enough to identify the final answer, because a candidate can satisfy the restriction and still fail the complete equation.
Step 2: Isolate the radical
Move terms that are outside the radical to the other side before squaring. For example, in √(x + 2) + 3 = 8, subtract 3 first:
√(x + 2) = 5
Step 3: Square both sides
Once the radical is alone, square each side:
(√(x + 2))² = 5²
This gives x + 2 = 25. Solve the simpler equation and then verify the candidate in the original form.
Restriction checklist
- For every even root, set its radicand greater than or equal to zero.
- Remember that a principal square root is never negative.
- If a square root is isolated and equals a number, that number must be nonnegative.
- If a denominator also contains a variable, exclude values that make the denominator zero.
- Use restrictions as an early filter, not as a substitute for checking.
Inputs and assumptions
Most introductory radical-equation problems assume that solutions are real numbers. Under that assumption, an expression such as √(x − 7) requires x − 7 ≥ 0. If a problem is specifically about complex numbers, the rules and interpretation change; use the conventions given in your course.
Keep these algebra assumptions in mind:
- Use parentheses when squaring: (2x − 1)² is not the same as 2x² − 1.
- Distribute only after writing the complete square: (a + b)² = a² + 2ab + b².
- Do not cancel across addition: a factor can be canceled only from a product or quotient, not from a sum.
- Use a calculator for checking, not for replacing the setup: enter the entire radicand with parentheses.
- Retain exact forms when possible: write √7 rather than rounding it during algebra.
If you need a refresher on moving terms and maintaining equality, review linear equations step by step. If squaring produces a quadratic, the methods in this quadratic-equation guide can help you finish the calculation.
Worked examples
Example 1: One square root and a linear answer
Solve √(x + 5) = 4.
- The radicand restriction is x + 5 ≥ 0, or x ≥ −5.
- The radical is already isolated.
- Square both sides: x + 5 = 16.
- Subtract 5: x = 11.
- Check: √(11 + 5) = √16 = 4.
The solution is x = 11.
Example 2: An extraneous solution
Solve √(x + 1) = x − 1.
- The radicand requires x + 1 ≥ 0. Also, because the left side is nonnegative, a valid right side must satisfy x − 1 ≥ 0, so a solution must have x ≥ 1.
- Square both sides: x + 1 = (x − 1)².
- Expand: x + 1 = x² − 2x + 1.
- Rearrange: 0 = x² − 3x.
- Factor: 0 = x(x − 3), giving candidates x = 0 and x = 3.
- Check both in the original equation. For x = 0, √1 = −1, which is false. For x = 3, √4 = 2, which is true.
Therefore, x = 3 is the solution, while x = 0 is extraneous.
Example 3: Two square roots
Solve √(x + 4) + √x = 4.
- The domain requires x ≥ 0.
- Isolate one radical: √(x + 4) = 4 − √x.
- Square both sides: x + 4 = 16 − 8√x + x.
- Cancel x and rearrange: 8√x = 12.
- Divide by 8: √x = 3/2.
- Square again: x = 9/4.
- Check: √(9/4 + 4) + √(9/4) = √(25/4) + 3/2 = 5/2 + 3/2 = 4.
The solution is x = 9/4. With two radicals, isolating one before each squaring step keeps the work organized and makes checking possible.
Practice set
Try these without looking at the answers first:
- √(x − 3) = 5
- √(2x + 1) − 2 = 4
- √(x + 6) = x
- √(x + 9) − √x = 1
Answers: 1) x = 28; 2) x = 25/2; 3) x = 3; 4) x = 16. Verify each answer in the original equation, not only in the equation created after squaring.
When to recalculate
Revisit your work whenever you change an input, copy a problem into a calculator, or move from one form of an equation to another. In a radical equation, the important “inputs” are the radicands, signs, coefficients, and any restrictions. A small change can alter the domain or create and remove candidate solutions.
Before submitting homework or using an equation solver, run this final routine:
- Rewrite the original equation clearly.
- List restrictions from even roots and denominators.
- Confirm that every radical was isolated before squaring.
- Check the expansion of every squared binomial.
- Substitute every candidate into the original equation.
- Round only at the end if a decimal answer is requested.
For extra practice, record whether each error came from a domain restriction, isolation step, expansion, or missed check. A simple mistake log makes the next set of equation practice problems more useful. If you are working with inequalities rather than equations, remember that the checking and sign rules differ; use an inequalities step-by-step guide instead.
The central habit is simple: isolate, square, solve, and check in the original equation. Follow that sequence every time, and radical equations become a repeatable algebra process rather than a guessing exercise.
