A radical equation is an equation in which a variable is under a radical. To solve a radical equation:
Isolate the radical expression involving the variable. If more than one radical expression involves the variable, then isolate one of them.
Raise both sides of the equation to the index of the radical.
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.