Use the Radical Equation Calculator when a square root, cube root, or other radical is part of your problem. It can help you simplify expressions, solve equations with variables inside radicals, and check whether a possible answer actually works.
How to Solve Radical Equations
First, decide what kind of task you have. Simplifying a radical expression means rewriting it in an equivalent, simpler form. Solving a radical equation means finding values of \(x\) that make the equation true.
When you solve, you usually isolate a radical, remove it by raising both sides to a matching power, and then verify every possible answer. That final check matters because squaring can create answers that were not valid before.
Simplify the Radical Before You Solve
Simplifying first can make an equation easier to read and solve. For square roots, look for perfect-square factors inside the radical.
\[ \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2} \]
You can combine radicals only when their simplified radical parts match:
\[ 3\sqrt{2}+5\sqrt{2}=8\sqrt{2} \]
For nonnegative real values of \(a\) and \(b\), multiply square roots using:
\[ \sqrt{a}\sqrt{b}=\sqrt{ab} \]
when the values stay in the real-number domain. But radicals do not distribute across addition:
\[ \sqrt{a+b}\ne\sqrt{a}+\sqrt{b} \]
Be careful with variables. In real-number work, an even root needs a nonnegative radicand. Also,
\[ \sqrt{x^2}=|x| \]
not always \(x\), because a square root cannot be negative. If you only need to rewrite an expression, a [Simplify Calculator](/simplify-calculator) can help with that task.
Isolate One Radical and Raise Both Sides
For an equation with one radical containing the variable, move everything else away from that radical first. Then raise both sides to the power that matches the radical index.
For a square root, square both sides:
\[ \sqrt{x+4}=7 \]
\[ x+4=49 \]
\[ x=45 \]
Keep the original equation visible while you work. After squaring and solving, substitute your answer back into the original equation to make sure it still works. This is one way to solve for \(x\) step by step when radicals are involved.
Solve Equations with More Than One Radical
When an equation has radicals on both sides, isolate one radical before you square. Simplify the result, then isolate the remaining radical and square again if needed.
That process can produce a quadratic equation after one or two squaring steps. If that happens, you can compare methods with the Quadratic Equation Calculator.
If a radical remains after the first squaring step, isolate that radical before squaring again. Each time you square, you may create more possible answers, so substitute every final candidate into the original equation.
Repeated squaring can produce possible values that do not satisfy the starting equation. Check every candidate. There is no real solution if an even-root radicand would need to be negative, or if every candidate fails the original equation.
Understanding Radical Equations
A radical sign reverses a power. For example,
\[ \sqrt{25}=5 \]
because \(5^2=25\). More generally, \(\sqrt[n]{a}=a^{1/n}\), which connects roots and rational exponents.
Notice that \(\sqrt{25}\) equals \(5\), not \(\pm5\). The radical symbol means the principal square root, which is nonnegative. The \(\pm\) sign belongs in a different situation, such as solving \(x^2=25\).
Squaring helps remove a square root, but it can make an incorrect value look acceptable. That is why checking the original equation is part of solving, not an optional extra step. You can find related practice tools under more Algebra tools.
Worked Example
Solve:
\[ \sqrt{x+5}=x-1 \]
Since the square root must be nonnegative, the right side must also be nonnegative.
\[ x-1\ge0 \]
\[ x\ge1 \]
This condition helps us recognize impossible candidates later.
Square both sides of the original equation.
\[ x+5=(x-1)^2 \]
The square root is removed, but we will still check answers in the original equation.
Expand and rearrange.
\[ x+5=x^2-2x+1 \]
\[ x^2-3x-4=0 \]
Now the equation is quadratic.
Factor to find possible values.
\[ (x-4)(x+1)=0 \]
\[ x=4 \quad \text{or} \quad x=-1 \]
These are candidates, not final answers yet.
Check \(x=4\) in the original equation.
\[ \sqrt{4+5}=4-1 \]
\[ 3=3 \]
So \(x=4\) works.
Check \(x=-1\) in the original equation.
\[ \sqrt{-1+5}=-1-1 \]
\[ 2\ne-2 \]
So \(x=-1\) is an extraneous solution created after squaring. The final answer is:
\[ x=4 \]
Questions
Questions Students Ask About Radical Equations
What is a radical equation?
+A radical equation has a variable inside a radical sign. For example, \(\sqrt{x+3}=5\) is a radical equation because \(x\) appears inside the square root. Solving means finding the value of \(x\) that makes both sides equal.
What is the difference between simplifying radicals and solving radical equations?
+Simplifying rewrites an expression in a cleaner but equivalent form, such as \(\sqrt{50}=5\sqrt2\). Solving finds the value or values of a variable that make an equation true. An expression may simplify without having an \(x\) to solve for.
How do I simplify a square root?
+Find the largest perfect-square factor inside the radicand. For example, \(\sqrt{48}=\sqrt{16\cdot3}=4\sqrt3\). If no perfect-square factor greater than 1 exists, the radical is already simplified over the real numbers.
Why do I square both sides of a radical equation?
+Squaring can remove a square root and turn the problem into a familiar algebra equation. For example, squaring \(\sqrt{x+1}=6\) gives \(x+1=36\). You must still check the result because squaring may introduce an extra candidate.
Why do I need to check my answer after squaring?
+If two expressions are equal, their squares are equal. But if two squares are equal, the original expressions may have different signs. That one-way relationship is why a value found after squaring may not satisfy the original equation.
Can a radical equation have no real solution?
+Yes. An even root cannot have a negative radicand in real-number work. An equation can also have no real solution when every candidate produced by algebra fails the original equation.
How do I multiply radical expressions?
+For appropriate real-number radicands, multiply inside one radical: \[ \sqrt{a}\cdot\sqrt{b}=\sqrt{ab} \] Then simplify the result if possible. For example, \(\sqrt6\cdot\sqrt{24}=\sqrt{144}=12\).
Why is \(\sqrt{x^2}\) equal to \(|x|\) instead of always \(x\)?
+The principal square root must be nonnegative. If \(x=-4\), then \(\sqrt{x^2}=\sqrt{16}=4\), not \(-4\). Absolute value gives the nonnegative result for both positive and negative values of \(x\).
What are extraneous solutions in a radical equation?
+An extraneous solution is a value that appears during the solving process but does not satisfy the original radical equation. Squaring both sides can introduce these false candidates, so always substitute each answer into the original equation.
How do I solve a radical equation with two radicals?
+Isolate one radical first, then square both sides to remove it. If another radical remains, isolate it and square again. Because repeated squaring can create extraneous solutions, check every final candidate in the original equation.
Can a radical equation have more than one solution?
+Yes. A radical equation can have more than one real solution, depending on its structure. Each solution must satisfy the original equation and all real-number restrictions.
How do I solve a cube-root equation?
+Isolate the cube root and then raise both sides to the third power. Unlike an even root, a cube root can have a negative radicand in real-number mathematics. For example, \(\sqrt[3]{x-2}=4\) gives \(x-2=64\), so \(x=66\).