13.2 - Radical equations
Before reading this section you may want to review the following topics:
A radical equation is one in which the unknown, call it x, is inside a radical.
To solve a radical equation follow these steps:
- Isolate the term containing the radical on one side of the equation,
say on the left-hand-side. (By isolate we mean get it by itself.)
- If the radical is a square root then square both sides of the equation.
(In general, if the radical is an nth root then take the
nth power of both sides.)
On the left-hand-side use the fact that
to get rid of the radical. On the right-hand-side distribute.
- The unknown x is no longer inside a radical.
Now you can finish solving for x
by using the basic procedures for solving equations.
Note:
- If the equation contains more than one radical term, you will have to
perform the above procedure several times.
- One of the steps, squaring both sides of the equation, increases the degree of
the unknown, and this often leads to extraneous solutions. Therefore it is
very important to check your solutions.
Example: Solve the radical equation 
.
Solution: The first step is to isolate the radical term:
Then square both sides:
6 x + 4 = 4 x 2.
The result is that this is no longer a radical equation; it is a quadratic equation.
Divide both sides by 2 to simplify it and then collect all terms on one side of the
equation to put it into standard form:
2 x 2 − 3 x − 2 = 0.
Click here to see how the left-hand-side of
this quadratic equation can be factored. The result is this:
(x − 2) (2 x + 1) = 0.
The purpose of factoring is to put the equation into the form
a · b = 0.
We can now replace it with two new equations. Each new equation comes from setting
one of the factors to zero:
x − 2 = 0
2 x + 1 = 0
Their solutions are x = 2 and x = −½.
We must check these solutions. Substituting x = 2 into the original
equation and simplifying causes the equation to read 0 = 0, so this solution checks out.
But substituting x = −½ back into the original equation
causes it to read 2 = 0, so this solution doesn’t check out and must be rejected.
Thus this radical equation has the single solution x = 2.
Example: Solve the radical equation 
.
Solution: The first step is to isolate one of the radical terms
(it doesn’t matter which one):
Then square both sides:
On the left-hand-side squaring got rid of the radical.
On the right-hand-side we will have to distribute.
After distributing and collecting like terms we get this equation:
This is still a radical equation because although one radical is gone,
the other one still remains. So we repeat the entire process.
Isolate the remaining radical term (by canceling x terms
and dividing by 32):
Then square both sides:
x − 32 = 49
Note that this is no longer a radical equation; both radicals are now gone.
Solving this equation yields the solution x = 81. We must now check the solution.
Substituting x = 81 back into the original equation and simplifying gives
the equation 16 = 16, so this solution checks out.
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