Math-3

Lesson 2-2Factoring Quadratics with Lead

Coefficient Not = 1,Complex Conjugates

Quiz: Factor the quadratic equation.

xxxy 42153 23

)4)(3)(2)(1( xxxxy

3x1x

2x4x

We are learning skills that will help us factor equations so that we can find x-intercepts.

)1)(6( xx

)1*6()16(2 xx

652 xx

What if there is not common factor but the lead coefficient is NOT equal to 1?

These come from multiplying binomials that also do not have lead coefficients of 1.

cbxaxy 2

)3)(12( xx

372 2 xx

Multiply this out (simplify)

)3)(12( xx

)3(1)3(2 xxx

Distributive property (twice)

3162 2 xxx

Notice something

372 2 xx

63*2

6*16 761

8143 2 xx

Multiply this out (simplify)

)4)(23( xx

)4(2)4(3 xxx

Distributive property (twice)

82123 2 xxx

Notice something

8143 2 xx

248*3

6*424

8*324

12*224

24*124

14122

What are the factors of 24?

15132 2 xx

Multiply this out (simplify)

)5)(32( xx

)5(3)5(2 xxx

Distributive property (twice)

153102 2 xxx

Notice something

15132 2 xx

3015*2

13310

3*1030

15132 2 xx15132 2 xx

)5)(32( xx

)5(3)5(2 xxx

Group the first two and last two terms

153102 2 xxx

This tells us to break

13x into 10x + 3x

3015*2

13310

3*1030 )153()102( 2 xxx

Factor out the common factors

Factor out the common factors

What are the factors of

30 that add up to 13?

10134 2 xx

)2)(54( xx

)2(5)2(4 xxx

Group the first two and last two terms

10584 2 xxx

This tells us to break

13x into 8x + 5x

4010*4

1358

5*840 )105()84( 2 xxx

Factor out the common factors

Factor out the common factors

10134 2 xx

8103 2 xx

)4)(23( xx

)23(4)23( xxx

Group the first two and last two terms

81223 2 xxx

This tells us to break

10x into -2x + 10x

248*3

10122

12*224 )812()23( 2 xxx

Factor out the common factors

Factor out the common factors

8103 2 xx

103 2 xx

)2)(53( xx

)2(5)2(3 xxx

Group the first two and last two terms

10563 2 xxx

This tells us to break

-x into -6x + 5x

3010*3

156

5*630 )105()63( 2 xxx

Factor out the common factors

Factor out the common factors

103 2 xx

352 2 xxy

Your turn: Factor the following quadratic equations.

9211 2 xxy

10139 2 xxy

51612 2 xxy

)1)(32( xx

)1)(911( xx

)2)(59( xx

)12)(56( xx

Your turn: factor the following binomials

62 xy

92 xy

)6)(6( xx

)9)(9( xx

)3)(3( xx

Imaginary Numbers

1

1x

Some number equals 1

22 1x

12 xWhy can’t this be a real number?

))(( ixix

“i” terms cancel.

)()( ixiixx

2x xi xi 2ii –squared = -1

2x )1(

12 x

Multiply this out:

))(( ixixy

How is this a transformation of the parent function?12 xy

Does it have any x-intercepts?

Does the equation have any values of ‘x’ that will cause ‘y’ to equal zero?

Yes: x = ‘i’ and ‘-i’

)2)(2( ixix

“i” terms cancel.

Multiply this out:

)2(2)2( ixiixx

2x 2xi 2xi )2*22ii –squared = -1

2x )2*2)1(

2x )2*2

22 x

Multiply out the 2 binomials to check it this is true.

?42 x )2)(2( ixix

What about the sum of two squares?

))((12 ixixx

22x

?32 x

)2)(2( ixix

)3)(3( ixix

)4)(4( ixix

))((2 aixaixax

Can you see the pattern?

Your turn: Factor the following equations

xxy 183 3 )9(3 2 xx

)3)(3(3 ixixx

xxy 255 3 )5(5 2 xx

)5)(5(5 ixixx

32 xy

)2)(2( xx

)3)(3( xx

22 xy

52 xy )5)(5( xx

172 xy )17)(17( xx

Your turn: Factor the following equations

1st TheoremIrrational Congjugates Theorem: IF an equation is of the form , where ‘a’ is not a perfect square, THEN it always factors into irrational conjugate pairs.

))((0 axax

ax 20

), aax

axy 2

1st TheoremComplex Congjugates Theorem: IF an equation is of the form , THEN it always factors into conjugate pairs of imaginary numbers.

))((0 aixaix

ax 20

), aiaix

axy 2