Chapter 9: Quadratic Functions 9.3 SIMPLIFYING …brownk/Math095/Math095.09.3.Simplifying...Simplify...
Transcript of Chapter 9: Quadratic Functions 9.3 SIMPLIFYING …brownk/Math095/Math095.09.3.Simplifying...Simplify...
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)
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