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