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 equation.

Isolate the radical expression.

equation

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

equation

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

Applying the quadratic formula, equation

Now, check the results.

If equation, equation

If x = –5,

equation

The solution is equation or x = –5.

Example 2

Solve equation.

Isolate the radical expression.

equation

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

Example 3

Solve equation.

Isolate one of the radical expressions.

equation

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

equation

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

equation

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

equation

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

Applying the quadratic formula, equation

Check the solutions.

If x = 10, equation

So x = 10 is not a solution.

If x = 2, equation

The only solution is x = 2.

Example 4

Solve equation.

Isolate the radical involving the variable.

equation

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.

equation

The check of the solution x = –15 is left to you.