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Chapter 9: Quadratic Functions
9.3 SIMPLIFYING RADICAL
EXPRESSIONS
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Vertex formula
ƒ f(
f(x)=Ax
) A 2+Bx+C
B C standard
t d d form
f
ƒ X coordinate of vertex is
ƒ Use this value in equation to find y
coordinate of vertex
ƒ ‘form’ is the way a function is written
ƒ ‘formula’ is a method to solve it
40
f(x)=2x
f(x)
2x2+10x+7
35
30
ƒ Graph
G h off th
the ffunction
ti
25
ƒ Can find vertex from
20
ffunction
i
ƒ Find axis of symmetry
using A and B
15
10
5
0
-10
-5
0
-5
-10
5
40
f(x)=2x
f(x)
2x2+10x+7
35
30
ƒ Vertex
V t fformula
l ffor x =
ƒ
25
=
20
15
ƒ Y=2((
ƒ Y=(
)2+10(( )+7
)
)-25+7=(
)-(
ƒ =
10
5
)
0
-10
-5
0
-5
-10
ƒ Vertex: (
,
)
5
Simplifying Radical Expressions
ƒ Radical:
R di l square roots
t and
d higher
hi h roots
t
ƒ Shorthand method of writing roots
à Use fractional exponent
à Not necessarily a fractional value of exponent
ƒ Some roots are ‘rational’
à Can be written as a ratio: exact value
ƒ Some roots are ‘irrational’
à Can only be written exact in ‘root’ form
Square Roots
ƒ Radical
R di l Sign:
Si
√
ƒ √9
ƒ
ƒ
ƒ
ƒ
number is radicand
Entire expression is the radical
Has a value: this one’s value is 3
3 is the principal square root of 9
The square roots of 9 include —3
3 also
√2x+5
ƒ Also
Al a radical
di l expression
i
ƒ Radicand is a binomial
à Composed of 2 terms
à Separated by addition
ƒ Important to recognize it is binomial,
not factors
√49
—√49
√49
√-49
√
49
ƒ Not
N t th
the same thi
thing!!
!!
ƒ First has principal sq.rt. of 7: 7·7=49
ƒ Second —7:
— (7)(7)
ƒ Third: not a real number
à There is not a real number you can multiply by
itself to get a negative product
ƒ When radicand is negative, there is not a
real square root
Irrational square roots: √48
ƒ Calculator
C l l t says 6.92820323
6 92820323
ƒ Not exact value!
ƒ We won’t use these approximations,
except to verify our simplified versions
ƒ We will learn method to simplify irrational
roots
Simplify square root: √36
ƒ √36
ƒ
ƒ
ƒ
ƒ
=6
36= 9 · 4
√36= √9 · 4 = √9
· √4
=3·2=6
Apply this method to irrational roots
Simplify square root: √48
ƒ √48 ≈ 6
6.92820323
92820323
ƒ 48= 16 · 3
ƒ √48= √16 · 3 =√16
· √3
ƒ = 4 · √3
ƒ Leave irrational part of root under radical
sign
g
Simplify square root: √72
ƒ √72
ƒ
ƒ
ƒ
ƒ
≈8
8.485281374
485281374
72= 36 · 2
√72= √36 · 2 =√36 · √2
= 6 · √2
Leave irrational part of root under radical
sign
g
Simplify square root: √72
ƒ √72
ƒ
ƒ
ƒ
ƒ
= √2
√2·2·3·3·2
2332
Prime factors of 72
√72= √ 2 · 2
√3·3 · √2
= 2·3√2
=6 √2
Square root of quotient
(division, fraction)
16 4
16
=
=
49 7
49
Square root of quotient
(division, fraction)
5
5
5
=
=
9
3
9
Not simplified,
simplified because a fraction
is under the radical sign
Square root of quotient
(division, fraction)
25⋅ 2 5 2
50
=
=
9⋅9
81
9
Not simplified,
simplified because a fraction
is under the radical sign
Square root of quotient
(division, fraction)
7
7 3
7 ⋅3
21
7
=
=
⋅
=
=
3
3
3 3
3⋅ 3
3
3
Not simplified,
simplified because a fraction
is under the radical sign
Also not simplified because there is
a radical in the denominator
Square root of quotient
(division, fraction)
3
3
3
5
3⋅5
15
=
=
⋅
=
=
20
4 ⋅ 5 2 5 5 2 5 ⋅ 5 10
Not simplified,
simplified because a fraction
is under the radical sign
Also not simplified because there is
a radical in the denominator
Simplifying a radical quotient
ƒ Note numerator is binomial
6+3 2 6 3 2 1
2 2
2 2+ 2
= +
= +
= +
=
12
12 12
2 4 4 4
4
ƒ FIRST write
it each
h term
t
off numerator
t over
the denominator!!
ƒ Reduce each fraction
ƒ Can then rewrite over single denominator
if you choose: doesn’t
doesn t really matter
Simplifying a radical quotient
8 − 28 8
28 4
4⋅7 4 2 7
= −
= −
= −
10
10 10
5
10
5 10
4
7 4− 7
= −
=
5 5
5
ƒ FIRST write each term of numerator over
the denominator!!
ƒ Reduce each fraction and simplify radical
ƒ Can then rewrite over single denominator
if you choose: doesn’t
doesn t really matter
Square Root Solutions for
Quadratic Equations
ƒ Section
S ti 9
9.4
4
ƒ Homework due on quiz day for chapter 9
ƒ November 3, next Wednesday
Use Square Root property
ƒ If you d
do th
the same thi
thing tto b
both
th sides
id off
equation, it is still a valid equation
ƒ Including
I l di taking
ki square root
ƒ Be sure to write ( ) around each side, so
you take the square root of the entire
side, not of separate terms on the side
√49
—√49
√49
√-49
√
49
ƒ Third:
Thi d nott a reall number
b
à There is not a real number you can multiply by
itself to get a negative product
ƒ When radicand is negative, there is not a
real square root
ƒ But radicand is -1·49…
ƒ Define
f
sq. rt. off -1 as “i“
“ “ ffor imaginary
i is the square root of —1
1
ƒ Factor
F t outt ffrom negative
ti radicands
di
d FIRST
ƒ The proceed to simplify the root
ƒ When solving equations, use both roots
ƒ ± sign:
g plus
p
or minus
ƒ Write +, then underline it with the —
ƒ If results has ± radical, ok to leave ±
ƒ If result is ± a value, add or subtract
value from the rest,
rest and get two answers