Math-2

Lesson 3-5

Review:

a) (3-1) Factoring Trinomials

b) (3-2) Solving by taking square roots

c) (3-3) Factoring Trinomials w/ lead coefficient ≠ 1

d) (3-4) Completing the square.

To Factor (verb) to break apart a number or an

expression into its factors.

= 2x + 6 2(x + 3)

distributive property: multiply terms that are being added.

To factor out the common factor: the “reverse” of the distributive property. creates a set of parentheses.

common factors can be numbers, variables, or a combination of the two.

Your turn: Factor out the common factor from each

binomial.

155 x

yyx 3311

)3(5 x

)3(11 xy

23 xx )1(2 xx

36 2024 xx )56(4 33 xx

x3

multiply )3)(2( xxThink: “left times left is the left term” 2x

)3)(2( xx Think: “right times right is the right term”

62x

)3)(2( xx Think: “inner”

62x x2

)3)(2( xx

62x x2

Think: “outer”

)3*2()32(2 xx

)3)(2( xx

)3*2()32(2 xx

652 xx

652 xx

Left times left is left

__)__)(__(__

__)__)(( xx Right times right is right

__)__)(( xx Right plus right is middle

)3)(2( xx What are the factors of 6

that add up to 5?

242 2 xx

)12(2 2 xx

Always factor out the

common factor first.

Now factor the trinomial.

)1)(1(2 xx

1662 xx )2)(8( xx

1892 xx )3)(6( xx

18246 2 xx )3)(1(6 xx

Find the “zeroes” of the equation.

)2)(1( xxy

y = 0

x = -1 x = +2

)4)(3( xxy 4 ,3 x

)10)(2(3 xxy 10 ,2x

12 x

Multiply the following two binomials:

)1)(1( xxy

When multiplying conjugate pairs of binomials, the coefficient of the ‘x’ term will be zero, resulting in another binomial.

12 xxxy

)2)(2( xxy

22222 xxxy 22 x

Finding the “Zeroes” by taking square roots.

42 xy

Set ‘y = 0” 40 2 x

Get the ‘x’ squared term by itself:

42 x

42 x

2x

“Zero” of a 2-variable equation: the input value that causes the output to equal zero.

But, “something squared” equals 4.

4) ( 2

42)( 2 42)( 2

2,2 x

Your turn: Find the zeroes.

122 xy

183 2 xy

“Isolate the square, undo the square”

814 2 xy

120 2 x212 x

x 12

32x

1830 2 x

2318 x26 x

6x

8140 2 x2481 x

6 ix

2

4

81x

x4

81

2

9x

“Nice” Quadratic Equations of the form: where ‘c’ is a negative number

32 xy

Always factors into:

162 xy

))(( mxmxy

)3)(3( xx

)16)(16( xx )4)(4( xx

cbxaxy 2

cxy 2

mxy 2

Or you can find the zeroes by taking square roots.

160 2 x 216 x x 4

30 2 x 23 x x 3

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

))((0 mxmx

mx 20

), mmx

mxy 2

2nd Theorem Complex Conjugates 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

162 xy )16)(16( ixix )4)(4( ixix

32 xy )3)(3( ixix

9)1(0 2 x

x 31

2)1(9 x

khxay 2)(

Vertex form extract a square root.

9)1( 2 xy

Isolate the squared term

Let y = 0

13 x

2,4 x

“Extract a square root”

Solve for ‘x’ simplify

2)1(9 x

“Isolate the square, undo the square”

16

Your turn: Find the “zeroes” by “Extracting a square root”

312 xy

522 xy

12432 xy

17722 xy

31x

52x

24x 6,2

2

177 x

2

347 x

103 2 xx

10563 2 xxx

This tells us to break

-x into -6x + 5x

3010*3

156

5*630

103 2 xx

What are the factors of -30 that add up to -1?

352 2 xxy

Break the following trinomials into 4 terms using the pattern we just learned.

654 2 xxy

485 2 xxy

3116 2 xxy

3322 2 xxxy

6384 2 xxxy

42105 2 xxxy

3296 2 xxxy

Vocabulary Factor by Grouping a method of factoring a 4-term polynomial

by grouping it into two groups of two then factoring each group

separately. 153102 2 xxxy

)153()102( 2 xxxy

Group the first two and last two terms

Factor out the common factor from each group separately.

)5(3)5(2 xxxyFactor out the common factor from each group (remember

this creates a set of parentheses).

)32)(5( xxy

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?

8143 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

14x into 2x + 12x

248*3

14122

12*224 )812()23( 2 xxx

Factor out the common factors

Factor out the common factors

8143 2 xx

What are the factors of 24 that add up to 14?

Always factor out the common

factor of all terms 1st.

)485(2 2 xxy Work inside the parentheses.

)]42()105[(2 2 xxxy “nested” parentheses.

)]2(2)2(5[2 xxxy

81610 2 xxy

Always rewrite an

equivalent form.

)2)(25(2 xxy

2x5

2 x

Find the zeroes.

Finding the zeroes of the polynomial by factoring.

8143 2 xxy

)4)(23(0 xx

32x

0)23( x 0)4( x

4 x

821230 2 xxx

)82()123(0 2 xxx

)4(2)4(30 xxx

Write this step out!!

(“1 step rewrite”)

Finding “zeroes” of unfactorable standard form quadratic equations.

cbxaxy 2

khxay 2)(

Complete the square

Turn standard form into vertex form, then solve directly.

(by extracting a square root)

(by completing the square)

x intercepts

Standard Form Quadratic

cbxaxy 2

Perfect Square Trinomial

2

2

21

bbxxy

2)1( xy

2)2( xy

2)3( xy

122 xxy

442 xxy

962 xxy

12

2

2

22

b

Vertex Form

22

)( bxxf

42

4

2

22

b

92

6

2

22

b

What is special about a Perfect square Trinomial?

Is a Binomial squared.

One x-intercept.

Vertex is x-intercept.

‘c’ is a perfect square. 2

2

2

bbxxy

Graph

Equation ‘a’ = 1

cbxaxy 2

2

2

bc

22

bxy

Completing the Square

2x

x5

2

105

2xx

x

x10

10

x55

x

x

25

25

x

x52x

2x x10

What must the number ‘c’

be equal to for it to be a

“perfect square trinomial”?

cxx 22C = ?

cxx 142C = ?

C = ? cxx 162

2

2

2

c 1

2

2

14

c 49

2

2

16

c 64

122 xx

49142 xx

64162 xx

Your turn:

64162 xxy

49142 xxy

2)8( xy

Rewrite the equation as the

square of a binomial:

2)7( xy

442 xxy

36122 xxy2)6( xy

2)2( xy

What form (standard form,

vertex form, or intercept form)

would you call the binomial

squared?

Your turn:

20102 xxy

khxay 2)(

What number would

complete the square?

2

2

10

25

Add and subtract 25 202525102 xxy

4525102 xxy Notice the perfect

square trinomial!!!

Convert to the square of a binomial 45)5( 2 xy

Vertex form!!!!!!

Rewrite in vertex form by

completing the square.

Find ‘zeroes’ by completing the square

(1) Convert to vertex form: 362 xxy

39962 xxy

(2) Set y = 0

(3) Isolate the square, undo the square

2)3(12 x

312 x

123x

123x

2

2

6

9

12)3( 2 xy

12)3(0 2 x2)3(12 x

123x

Solving by completing the square (1) Convert to vertex form: 4100 2 xx

42525102 xxy

(2) Set y = 0

(3) Isolate the square, undo the square

2)5(29 x

529 x

295x

295x

2

2

10

25

29)5(0 2 x

29)5(0 2 x2)5(29 x

295x

162 xxy

52162 xxy

8)3( 2 xy

12)8( 2 xy

Your turn: Rewrite in vertex form by

completing the square.

622 xxy

1362 xxy

71 ,71 x

iix 23 ,23

Find the zeroes by completing the square.