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TEKS
4.5
2A.6.A, 2A.6.B,
2A.8.A, 2A.8.D
Key Vocabulary
• square root
• radical
• radicand
• rationalizing the
denominator
• conjugates
Before You solved quadratic equations by factoring.
Now You will solve quadratic equations by finding square roots.
Why? So you can solve problems about astronomy, as in Ex. 39.
USE A CALCULATOR
You can use a calculator
to approximate Ï }
s
when s is not a perfect
square. For example,
Ï }
80 ø 8.944.
Solve Quadratic Equations by
Finding Square Roots
A number r is a square root of a number s if r 2 5 s. A positive number s has two
square roots, written as Ï }
s and 2Ï }
s. For example, because 3 2 5 9 and (23) 2 5 9,
the two square roots of 9 are Ï }
9 5 3 and 2Ï }
9 523. The positive square root of a
number is also called the principal square root.
The expression Ï }
s is called a radical. The symbol Ï }
is a radical sign, and the
number s beneath the radical sign is the radicand of the expression.
E XAMPLE 1 Use properties of square roots
Simplify the expression.
266 Chapter 4 Quadratic Functions and Factoring
KEY CONCEPT For Your Notebook
Properties of Square Roots (a > 0, b > 0)
Product Property Ï }
ab 5 Ï }
a p Ï }
b Example Ï }
18 5 Ï }
9 p Ï }
2 5 3Ï }
2
Quotient Property Î } a
}b 5 Ï} a
}
b
a. Ï }
80 5 Ï }
16 p Ï }
5 5 4Ï }
5 b. Ï }
6 p Ï }
21 5 Ï }
126 5 Ï }
9 p Ï }
14 5 3Ï }
14
c. Î }
4
} 5
81 Ï} 4
} Ï 81 5 2 }
9
Ï }
Example Î }
2
} 5
25 Ï} 2
} Ï 25
d. Î }
7
} 5
16 Ï} 7
} Ï 16
Ï}
5
7
}
4
Ï}
5
2
}
5
SIMPLIFYING SQUARE ROOTS You can use the properties above to simplify
expressions containing square roots. A square-root expression is simplified if:
• no radicand has a perfect-square factor other than 1, and
• there is no radical in a denominator
✓ GUIDED PRACTICE for Example 1
Simplify the expression.
1. Ï }
27 2. Ï }
98 3. Ï }
10 p Ï }
15 4. Ï }
8 p Ï }
28
5. Î }
9
}
64
6. Î }
15
}
4
7. Î }
11
}
25
8. Î }
36
}
49