)4)(3)(2)(1( xxxxy
3x1x
2x4x
We are learning skills that will help us factor equations so that we can find x-intercepts.
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
))(( 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