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LESSON 6
Solving Radical Equations
1
Solve Radical Equations
Radical equations
include radical expressions. You can
solve a radical equation by raising each side of the equation to a power.
Key Concept
Solving Radical Equations
Step 1
Isolate the radical on one side of the equation.
Step 2
Raise each side of the equation to a power equal to the index of the radical to eliminate
the radical.
Step 3
Solve the resulting polynomial equation. Check your results.
When solving radical equations, the result may be a number that does not satisfy the
original equation. Such a number is called an
extraneous solution
.
Example 1
Solve Radical Equations
Solve each equation.
a.
√
x
+
2
+
4
=
7
√
x
+
2
+
4
=
7
Original equation
√
x
+
2
=
3
Subtract 4 from each side to isolate the radical.
(
√
x
+
2
)
2
=
3
2
Square each side to eliminate the radical.
x
+
2
=
9
Find the squares.
x
=
7
Subtract 2 from each side.
CHECK
√
x
+
2
+
4
=
7
Original equation
√
7
+
2
+
4
≟
7
Replace
x
with 7.
7
=
7
✓
Simplify.
b.
√
x
-
12
=
2
-
√
x
√
x
-
12
=
2
-
√
x
Original equation
(
√
x
-
12
)
2
=
(
2
-
√
x
)
2
Square each side.
x
-
12
=
4
-
4
√
x
+
x
Find the squares.
-
16
= -
4
√
x
Isolate the radical.
4
=
√
x
Divide each side by
-
4.
16
=
x
Square each side.
Why?
When you jump, the time that you are in the air is
your hang time. Hang time can be calculated in
seconds
t
if you know the height
h
of the jump in
feet. The formula for hang time is
t
=
0.5
√
h
.
A volleyball player had a hang time of about 0.67 second.
How would you calculate the height of her jump?
Now
1
Solve equations
containing radicals.
2
Solve inequalities
containing radicals.
Then
You solved
polynomial
equations.
New
Vocabulary
radical equation
extraneous solution
radical inequality
Mathematical
Practices
4
Model with
mathematics.
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352
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Lesson 5-6