9.3 SIMPLIFYING RADICAL Chapter 9: Quadratic Functions

EXPRESSIONS

Vertex formulaVertex formula

f( ) A 2 B C t d d ff(x)=Ax2+Bx+C standard formX coordinate of vertex isUse this value in equation to find y coordinate of vertex‘form’ is the way a function is written‘formula’ is a method to solve it

f(x)=2x2+10x+7 35

40

f(x) 2x +10x+7

G h f th f ti 25

30

35

Graph of the functionCan find vertex fromf i 15

20

25

functionFind axis of symmetry

10

15

using A and B0

5

-10 -5 0 5

-10

-5

f(x)=2x2+10x+7 35

40

f(x) 2x +10x+7

V t f l f 25

30

35

Vertex formula for x ==

15

20

25

Y=2( )2+10( )+710

15

( ) ( )Y=( )-25+7=( )-( )=

0

5

-10 -5 0 5

Vertex: ( )-10

-5

Vertex: ( , )

Simplifying Radical ExpressionsSimplifying Radical Expressions

R di l t d hi h tRadical: square roots and higher rootsShorthand method of writing roots

Use fractional exponentNot necessarily a fractional value of exponent

Some roots are ‘rational’Can be written as a ratio: exact value

Some roots are ‘irrational’Can only be written exact in ‘root’ form

Square RootsSquare Roots

R di l Si √Radical Sign: √√9 number is radicandEntire expression is the radicalHas a value: this one’s value is 33 is the principal square root of 9The square roots of 9 include —3 alsoThe square roots of 9 include 3 also

√2x+5√2x+5

Al di l iAlso a radical expressionRadicand is a binomial

Composed of 2 termsSeparated by addition

Important to recognize it is binomial, not factors

√49 —√49 √-49√49 √49 √ 49

N t th thi !!Not the same thing!!First has principal sq.rt. of 7: 7·7=49Second —7: — (7)(7)Third: not a real number

There is not a real number you can multiply by itself to get a negative product

When radicand is negative, there is not a real square root

Irrational square roots: √48Irrational square roots: √48

C l l t 6 92820323Calculator says 6.92820323Not exact value!We won’t use these approximations, except to verify our simplified versionsWe will learn method to simplify irrational roots

Simplify square root: √36Simplify square root: √36

√36 6√36 = 636= 9 · 4√36= √9 · 4 = √9 · √4= 3 · 2 = 6Apply this method to irrational roots

Simplify square root: √48Simplify square root: √48

√48 6 92820323√48 ≈ 6.9282032348= 16 · 3√48= √16 · 3 =√16 · √3= 4 · √3Leave irrational part of root under radical signg

Simplify square root: √72Simplify square root: √72

√72 8 485281374√72 ≈ 8.48528137472= 36 · 2√72= √36 · 2 =√36 · √2= 6 · √2Leave irrational part of root under radical signg

Simplify square root: √72Simplify square root: √72

√72 √2 2 3 3 2√72 = √2·2·3·3·2Prime factors of 72√72= √ 2 · 2 √3·3 · √2= 2·3√2=6 √2

Square root of quotientSquare root of quotient (division, fraction)

44916

4916

=74

=49 49 7

Square root of quotientSquare root of quotient (division, fraction)

95

95

=35

=9 9 3

Not simplified because a fractionNot simplified, because a fraction is under the radical sign

Square root of quotientSquare root of quotient (division, fraction)

8150

99225 ⋅

=925

=81 99 ⋅ 9

Not simplified because a fractionNot simplified, because a fraction is under the radical sign

Square root of quotientSquare root of quotient (division, fraction)

37

37

=33

37⋅=

321

3337=

⋅=

3 3

Not simplified because a fraction

33 333 ⋅

Not simplified, because a fraction is under the radical sign

Also not simplified because there is a radical in the denominatora radical in the denominator

Square root of quotientSquare root of quotient (division, fraction)

203

543

=55

523⋅=

1015

55253

=⋅

=20 54 ⋅

Not simplified because a fraction

552 10552 ⋅

Not simplified, because a fraction is under the radical sign

Also not simplified because there is a radical in the denominatora radical in the denominator

Simplifying a radical quotientSimplifying a radical quotient

Note numerator is binomialNote numerator is binomial

236+ 236 21 2222 +

FIRST it h t f t

12236+

1223

126+=

42

21+=

422

42

42 +

=+=

FIRST write each term of numerator over the denominator!!Reduce each fractionReduce each fractionCan then rewrite over single denominator if you choose: doesn’t really matterif you choose: doesn t really matter

Simplifying a radical quotientSimplifying a radical quotient288− 288

=744 ⋅

=724

=10 1010

−=105

−=105

−=

7474 −

FIRST write each term of numerator over5

757

5=−=

FIRST write each term of numerator over the denominator!!Reduce each fraction and simplify radicalReduce each fraction and simplify radicalCan then rewrite over single denominator if you choose: doesn’t really matterif you choose: doesn t really matter

Square Root Solutions forSquare Root Solutions for Quadratic Equations

S ti 9 4Section 9.4Homework due on quiz day for chapter 9November 3, next Wednesday

Use Square Root propertyUse Square Root property

If d th thi t b th id fIf you do the same thing to both sides of equation, it is still a valid equationI l di kiIncluding taking square rootBe 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 √ 49

Thi d t l bThird: not a real numberThere is not a real number you can multiply by itself to get a negative productitself to get a negative product

When radicand is negative, there is not a real square rootreal square rootBut radicand is -1·49…

f f “ “ fDefine sq. rt. of -1 as “i“ for imaginary

i is the square root of —1i is the square root of 1

F t t f ti di d FIRSTFactor out from negative radicands FIRSTThe proceed to simplify the rootWhen solving equations, use both roots± sign: plus or minusg pWrite +, then underline it with the —If results has ± radical, ok to leave ±If results has ± radical, ok to leave ±If result is ± a value, add or subtract value from the rest and get two answersvalue from the rest, and get two answers