Solving Radical Equations

A radical equation is an equation in which a variable is under a radical. To solve a radical equation:

1. Isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them.

2. Raise both sides of the equation to the index of the radical.

3. If there is still a radical equation, repeat Steps 1 and 2; otherwise, solve the resulting equation and check the answer in the original equation .

By raising both sides of an equation to a power, some solutions may have been introduced that do not make the original equation true. These solutions are called extraneous solutions.

Example 1: Solve

Isolate the radical expression.

Raise both sides to the index of the radical; in this case, square both sides.

This quadratic equation can now be solved either by factoring or by applying the quadratic formula.

Applying the quadratic formula,

Now, check the results.

If

If ,

The solution is or

Example 2: Solve 7

Isolate the radical expression.

There is no solution, since cannot have a negative value.

Example 3: Solve

Isolate one of the radical expressions.

Raise both sides to the index of the radical; in this case, square both sides.

This is still a radical equation. Isolate the radical expression.

Raise both sides to the index of the radical; in this case, square both sides.

This can be solved either by factoring or by applying the quadratic formula.

Applying the quadratic formula,

Check the solutions.

If ,

So is not a solution.

If ,

The only solution is .

Example 4: Solve

Isolate the radical involving the variable.

Since radicals with odd indexes can have negative answers, this problem does have solutions. Raise both sides of the equation to the index of the radical; in this case, cube both sides.

The check of the solution is left to you.

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