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Hawkes Learning Systems:
College Algebra
Section 2.6: Radical Equations
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Objectives
o Solving radical equations.
o Solving equations with positive rational exponents.
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Radical Equations
o A radical equation is an equation that has at least
one radical expression containing a variable, while
any non-radical expressions are polynomial terms.
o One reasonable approach to solving these equations
is to raise both sides of the equation to whatever
power is necessary to “undo” the radical(s).
o This may result in some extraneous solutions that we
must identify and discard by checking all of our
eventual solutions in the original equation.
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Radical Functions
For example, consider the equation
x  4
x 2  16
x   16
Square both sides.
Square root both sides.
From this, we obtain the solution that x  4 or x  4 .
We have gained a second (and false) solution.
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Solving Radical Equations
Method of Solving Radical Equations
Step 1: Begin by isolating the radical expression on one
side of the equation. If there is more than one
radical expression, choose one of the radical
expressions to isolate on one side.
Step 2: Raise both sides of the equation by the power
necessary to “undo” the isolated radical. That is,
if the radical is an nth root, raise both sides to
th
n
the
power.
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Solving Radical Equations
Method of Solving Radical Equations (Cont.)
Step 3: If any radical expressions remain, simplify the
equation if possible and then repeat steps 1 and
2 until the result is a polynomial equation.
When a polynomial equation has been
obtained, solve the equation using polynomial
methods.
Step 4: Check your solutions in the original equation.
Any extraneous solutions must be discarded.
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Example 1: Solving Radical Equations
2 x 2  x
Solve the radical equation.
2 x  x2
Step 1: Isolate the radical
expression.
Step 2: Square both sides.
Step 3: Solve the
polynomial equation.
Note: Both roots satisfy the
original equation.

2 x

2
  x  2
2
2  x  x2  4x  4
0  x 2  3x  2
0   x  1 x  2 
x  1, 2
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Example 2: Solving Radical Equations
Solve the radical equation.
x  3  x  4 1

x3
  1 
2
x4

2
x  3  1 2 x  4  x  4
2 x4 2

x4

2
 1
x  4 1
x  3
2
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Example 3: Solving Radical Equations
Solve the radical equation.
4

x2  x  1  1  0
4

4
x  x  1  1
2
4
x2  x  1  1
x2  x  2  0
 x  2 x  1  0
x  2, 1
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Example 4: Solving Radical Equations
Solve the radical equation.

1  x 1  x
1 x  x 1
1 x

2
  x  1
2
1  x  x2  2x  1
0  x  3x
2
0  x  x  3
x  0, 3
x0
Note that 1  (3)  (3)  1, so -3 is
an extraneous solution.
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Solving Radical Equations
Caution!
Substitute solutions back into the original equation to
check them! Some solutions may not fit, and are
therefore extraneous.
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Solving Equations with Positive Rational
Exponents
o Equations containing terms with positive rational
exponents can be viewed as radical equations.
o Rewriting each term that has a positive rational
exponent as a radical will allow us to use the method
developed previously to solve rational equations.
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Example 4: Equations with Positive Rational
Exponents
Solve the following equation with a rational exponent.
2
3
Step 1: Isolate the term containing
the rational exponent.
x  16  0
2
3
x  16
3
Step 2: Cube both sides to eliminate
the cubed root.
Step 3: Take the square root of both
sides.
Step 4: Verify that both numbers
satisfy the original equation.
x 2  16
x 2  163


x   16 


1
2
x   4 
x  64
3
3
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Example 5: Equations with Positive Rational
Exponents
Solve the following equation with a rational exponent.
1
4
(7 x 2  21x  130)  4
4
7 x 2  21x  130  4
7 x2  21x  130  44
7 x2  21x  130  256
7 x2  21x 126  0
7  x 2  3 x  18   0
x 2  3x  18  0
 x  3 x  6  0
x  3,6