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: Note:


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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