**Radical equations** are equations that contain variables in the **radicand** (the expression under a radical symbol), such as

Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find **extraneous solutions**, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

### A General Note: Radical Equations

An equation containing terms with a variable in the radicand is called a **radical equation**.

### How To: Given a radical equation, solve it.

- Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.
- If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an
*n*th root radical, raise both sides to the*n*th power. Doing so eliminates the radical symbol. - Solve the remaining equation.
- If a radical term still remains, repeat steps 1–2.
- Confirm solutions by substituting them into the original equation.

### Example 6: Solving an Equation with One Radical

Solve [latex]\sqrt{15 - 2x}=x[/latex].

### Solution

The radical is already isolated on the left side of the equal side, so proceed to square both sides.

We see that the remaining equation is a quadratic. Set it equal to zero and solve.

The proposed solutions are [latex]x=-5[/latex] and [latex]x=3[/latex]. Let us check each solution back in the original equation. First, check [latex]x=-5[/latex].

This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.

Check [latex]x=3[/latex].

The solution is [latex]x=3[/latex].

### Example 7: Solving a Radical Equation Containing Two Radicals

Solve [latex]\sqrt{2x+3}+\sqrt{x - 2}=4[/latex].

### Solution

As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.

Use the perfect square formula to expand the right side: [latex]{\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}[/latex].

Now that both radicals have been eliminated, set the quadratic equal to zero and solve.

The proposed solutions are [latex]x=3[/latex] and [latex]x=83[/latex]. Check each solution in the original equation.

One solution is [latex]x=3[/latex].

Check [latex]x=83[/latex].

The only solution is [latex]x=3[/latex]. We see that [latex]x=83[/latex] is an extraneous solution.