Section 6.5
Radical Equations
A radical equation is an equation where the variable that we are solving for is under the radical symbol.
Let’s start with a few very easy examples:
Example 1:
√𝑥 = 5
Just by looking we can tell that the answer would be x = ______________
Now, let’s check our answer in the original.
Example 2:
√𝑥 = −5
When we take the square root of a number the answer must be positive.
So is it possible to take the square root of a number and get an answer of -5? ________
So the answer must be ___________________ .
Steps to solve a radical equation:
1. Isolate the radical expression.
2. Get rid of the radical by raising both sides to the appropriate power.
3. Solve the resulting equation.
4. Check each potential solution in the original equation. If it does not work in the original equation, then we
call that solution extraneous.
Let’s use the steps on the equations from Example 1.
√𝑥 = 5
Example 3: Use the four steps to solve the following radical equations for the given variable.
a)
c)
√4𝑥 + 1 − 5 = 0
√𝑎 + 10 = 𝑎 − 2
b) √𝑥 2 + 16 + 6 = 1
√3𝑥 + 4 − 2 = 𝑥
d)
3
e) √3𝑎 + 5 = −3
4
g) √3𝑥 + 1 = 2
f)
3
3
√3𝑥 + 5 = √5 − 2𝑥