Download Solving Quadratic Equations by Finding Square Roots

5.3
Solving Quadratic Equations by
Finding Square Roots
Goals p Solve quadratic equations.
p Use quadratic equations to solve real-life problems.
Your Notes
VOCABULARY
Square root A number r is a square root of a number s
if r 2 ! s.
Radical sign The symbol
root
!3", which denotes a square
Radicand The number or expression beneath a radical
sign
Radical An expression of the form
number or expression
!s" where s is a
Rationalizing the denominator The process of
eliminating a radical in the denominator of a fraction
by multiplying both the numerator and the
denominator by an appropriate radical
PROPERTIES OF SQUARE ROOTS (a > 0, b > 0)
Product Property: !ab
"!
Quotient Property:
#"$ab" !
!a" p !b
"
!a"
!b
"
Lesson 5.3 • Algebra 2 Notetaking Guide
101
Your Notes
Using Properties of Square Roots
Example 1
Simplify the expression.
!9" p !3" ! 3!3"
a. !27
"!
b. !5
" p !15
"!
c.
5
"!
#"$
36
d.
#$
13
"" !
3
!75
" ! !25
" p !3" ! 5!3"
!5"
!36
"
!"
13
!"
5
! ""
6
p
!3"
!3"
!3"
!39
"
! ""
3
Checkpoint Simplify the expression.
1. !5
" p !8
"
2.
!"
15
""
5
2!"
10
Example 2
#$"35"
Solving a Quadratic Equation
1
Solve ""(x # 2)2 ! 8.
2
Solution
1
""(x # 2)2 ! 8
2
(x #2)2 ! 16
x # 2 ! $!"
16
x ! 2$4
Write original equation.
Multiply each side by 2 .
Take square roots of each side.
Add 2 to each side.
The solutions are #2 and 6 .
Check Check the solutions either by substituting them into
1
the original equation or by graphing y ! ""(x # 2)2 # 8
2
and observing the x-intercepts.
102
Algebra 2 Notetaking Guide • Chapter 5
Your Notes
Example 3
Modeling a Falling Object’s Height
An acorn falls out of a tree from a height of 40 feet. How
many seconds does the acorn take to reach the ground?
Solution
The initial height is h0 ! 40 feet, so the height as a function
of time is h ! #16t2 % 40. Find the value of t for which
h ! 0 to determine how long it takes the acorn to reach the
ground.
Method 1: Make a table of values.
t
0
The table shows that h ! 0 has a
h
40
value of t between t ! 1 and t ! 2 .
It takes between 1 second and
2 seconds for the acorn to reach the ground.
1
2
24
#24
Method 2: Solve a quadratic equation.
h ! #16t2 % 40
0 ! #16t2 % 40
#40 ! #16t2
2.5 ! t2
!2.5
"!t
1.6 ≈ t
Write height function.
Substitute 0 for h.
Subtract 40 from each side.
Divide each side by #16 .
Take positive square root.
Use a calculator.
The acorn takes about 1.6 seconds to reach the ground.
Checkpoint Complete the following exercises.
3. Solve 5x2 # 30 ! 70.
"
$2!5
Homework
1
4. Solve ""(x % 7)2 ! 8.
3
#7 $ 2!6
"
5. You drop a football from a window 20 feet above the
ground. For how much time is the football falling if your
friend catches it at a height of 4 feet?
1 second
Lesson 5.3 • Algebra 2 Notetaking Guide
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