factoring quadratic trinomials …beyond the guess and test method

Factoring Quadratic Trinomials

…beyond the guess and test method.

Topics

1. Standard Form2. When c is positive and b is positive3. When c is positive and b is negative4. When c is negative5. When the trinomial is not factorable6. When a does not equal 17. When there is a GCF

The standard form of any quadratic trinomial is

Standard Form

cbxax ++2

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So, in 3x 2 − 4x + 1... a=3

b=-4

c=1

Now you try.

272 −+− xx

a = ??

b = ??c = ??

Click here when you are ready to check your answers!

The standard form of any quadratic trinomial is

Recall

cbxax ++2

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So, in 2x 2 − x + 5... a = 2

b = -1

c = 5

Try Another!

24 2 −+ xxa = ??

b = ??c = ??

Go on to factoring!

Factoring when a=1 and c > 0.

First list all the factors of c.

€

x 2 + 8x + 121 12

2 6

3 4

Find the pair that adds to ‘b’

1 12

2 6

3 4

These numbers are used in the factored expression.

€

x + 2( ) x + 6( )

Now you try.

€

x 2 + 9x + 20€

x 2 + 8x + 15

€

x 2 + 10x + 21


1.

2.

3.

Recall

€

x 2 + 10x + 24

1 24

2 12

3 8

4 6So we get:

€

x + 4( ) x + 6( )

Try some others!

We need to list the factors of c.

96.1 2 ++ xx

(x+3)(x+3) 78.2 2 ++ xx(x+2)(x+3)

67.2 2 ++ xx

(x+1)(x+6) 78.2 2 ++ xx(x+2)(x+3)

Go on to factoring where b is negative!

Factoring when c >0 and b < 0.

Since a negative number times a negative number produces a positive answer, we can use the same method.

Just remember to use negatives in the expression!

Let’s look at

€

x 2 − 13x + 12

1 12

2 6

3 4

€

x − 12( ) x − 1( )

We need a sum of -13

Make sure both values are negative!

First list the factors of 12

Now you try.

€

x 2 − 5x + 4

x 2 − 9x + 14

x 2 − 13x + 42


1.

2.

3.

Recall 862 +− xx

1 8

2 4

In this case, one factor should be positive and the other negative.

€

x − 2( ) x − 4( )

We need a sum of -6

Try some others!

127.1 2 +− xx

(x-3)(x-4) 78.2 2 ++ xx(x-3)(x+4)

44.2 2 +− xx

(x-2)(x-2) 78.2 2 ++ xx(x-1)(x-4)

Go on to factoring where c is negative!

Factoring when c < 0.

We still look for the factors of c. However, in this case, one factor should be positive

and the other negative.

Remember that the only way we can multiply two numbers and

come up with a negative answer, is if one is number is positive and the other is negative!

Let’s look at

€

x 2 − x − 12

1 12

2 6

3 4


€

x + 3( ) x − 4( )

We need a sum of -1

Now you try.

€

x 2 + 3x − 4

x 2 + x − 20

x 2 − 4x − 21

x 2 − 10x − 56


1.

2.

3.

4.

Recall

€

x 2 + 3x − 18

1 18

2 12

3 6


€

x − 3( ) x + 6( )We need a sum of 3

Try some others!

€

1. x 2 − 2x −15

(x-3)(x+5) 78.2 2 ++ xx(x+3)(x-5)

30.2 2 −+ xx

(x-5)(x+6) 78.2 2 ++ xx(x-6)(x-5)

Go on to trinomials that are not factorable

Prime Trinomials

Sometimes you will find a quadratic trinomial that is not

factorable. You will know this when

you cannot get b from the list of factors.

When you encounter this write not factorable or

prime.

Here is an example…

€

x 2 + 3x + 18 1 18

2 9

3 6

Since none of the pairs adds to 3, this trinomial is prime.

Now you try.

€

x 2 − 6x + 4

€

x 2 − 10x − 39

€

x 2 + 5x − 7

factorable prime

factorable prime

factorable prime

Go on to factoring when a≠1

When a ≠ 1.

Instead of finding the factors of c:Multiply a times c.Then find the factors of this product.

€

7x 2 − 19x + 10

a × c = 70

1 70

2 35

5 14

7 10

1 70

2 35

5 14

7 10

We still determine the factors that add to b.

So now we have

But we’re not finished yet….

€

x − 5( ) x − 14( )

Since we multiplied in the beginning, we need to divide in the end.

€

x −5

7

⎛

⎝ ⎜

⎞

⎠ ⎟ x −

14

7

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x −5

7

⎛

⎝ ⎜

⎞

⎠ ⎟ x − 2( )

Divide each constant by a.Simplify, if possible.

€

7x − 5( ) x − 2( )Clear the fraction in each binomial factor

Recall

Divide each constant by a.

Simplify, if possible.

Clear the fractions in each factor

Multiply a times c.

List factors.

Write 2 binomials with the factors that add to b

932 2 −− xx

1892 =×

63

92

181

( )( )36 +− xx

⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ −23

26

xx

( ) ⎟⎠

⎞⎜⎝

⎛ +−23

3 xx

( )( )323 +− xx

Try some others!

Now you try.

7236.3

1253.2

344.1

2

2

2

+−

−−

−+

xx

xx

xx


592.1 2 −+ xx

(2x-1)(x+5) 78.2 2 ++ xx(2x+5)(x+1)

564.2 2 −− xx

(2x-5)(2x+1) 78.2 2 ++ xx(4x+5)(x-1)

Go on to trinomials that have a GCF

Sometimes there is a GCF.

If so, factor it out first.

3024x Ex) 2 −− x

€

2 2x 2 − x − 15( )

€

2 × 15 = 301 302 153 105 6

€

2 x − 6( ) x + 5( )

€

2 x −6

2

⎛

⎝ ⎜

⎞

⎠ ⎟ x −

5

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

2 x − 3( ) x −5

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

2 x − 3( ) 5x + 2( )

Now you try.

€

4x 2 + 16x + 12

6x 2 + 10x + 6


1.

2.

Recall

First factor out the GCF.

103545x 2 −− x

( )2795 2 −− xx

118

2 93 6

( )( )925 −+ xx

⎟⎠

⎞⎜⎝

⎛ −⎟⎠

⎞⎜⎝

⎛ +99

92

5 xx

( )( )1295 −+ xx

( )( )1295 −+ xx

Then factor the remaining trinomial.

9 times 2 = 18

Try some others!

€

6x 2 + 30x − 36

6(x-1)(x+6) 78.2 2 ++ xx(6x+6)(x-6)

€

4x 2 + 14x + 10

2(2x+1)(x+5) 78.2 2 ++ xx2(2x+5)(x+1)

1.

2.

Did you get these answers?

2

7

1

−==−=

cba

Yes No


( )( )45.3 ++ xx

( )( )53.1 ++ xx

( )( )73.2 ++ xx

Yes No


( )( )76.3 −− xx

( )( )41.1 −− xx

( )( )72.2 −− xx

Yes No


( )( )( )( )( )( )( )( )144.4

73.3

54.2

41.1

−+

−+

+−

+−

xx

xx

xx

xx

Yes No


( )( )( )( )( )( )7213.3

433.2

3212.1

−−

+−

+−

xx

xx

xx

Yes No


( )( )prime.2

314.1 ++ xx

Yes No

Good Job!You have completed

Standard Form!


factoring “When c is positive and b is

positive”!


factoring “When c is positive and b is negative”!

Good Job!

Good Job! You have completed

factoring “When a does not equal 1”!


factoring“When c is negative”!

Good Job!


factoring “When there is a GCF”!

Review and Try Again!

Try Again!

factoring quadratic trinomials …beyond the guess and test method

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