## Radical Equations

This is a refresher on solving various radical equations and determining whether there are extraneous solutions. Radical is another term for root, so you will be solving equations that contain square roots.

### Solving Radical Equations

**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.

#### 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.

1. Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.

2. 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 th root radical, raise both sides to the th power. Doing so eliminates the radical symbol.

3. Solve the remaining equation.

4. If a radical term still remains, repeat steps 1-2.

5. Confirm solutions by substituting them into the original equation.

#### EXAMPLE 6

##### Solving an Equation with One Radical

##### 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 and . Let us check each solution back in the original equation. First, check .

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

#### TRY IT #5

#### EXAMPLE 7

##### Solving a Radical Equation Containing Two Radicals

##### 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: .

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

The proposed solutions are and . Check each solution in the original equation.

The only solution is . We see that is an extraneous solution.

#### TRY IT #6

Solve the equation with two radicals: .

Source: Rice University, https://openstax.org/books/college-algebra/pages/2-6-other-types-of-equations

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