Unlocking the Secrets of Factoring Quadratic Expressions with Greatest Common Factor and Zero Product Property

Factors of $3$	Sum of the Factors
$- 1$ and $- 3$	$- 1 + (- 3) = - 4 \times$
$1$ and $3$	$1 + 3 = 4 ✓$

Factors of

3

Sum of the Factors

- 1

and

- 3

- 1 + (- 3) = - 4 \times

1

and

3

1 + 3 = 4 ✓

	Substitution	Calculation
$w = x + 3$	$w = 1 + 3$	$w = 4$ feet
$ℓ = 3 x + 3$	$ℓ = 3 \cdot 1 + 3$	$ℓ = 6$ feet

Substitution

Calculation

w = x + 3

w = 1 + 3

w = 4

feet

ℓ = 3 x + 3

ℓ = 3 \cdot 1 + 3

ℓ = 6

feet

Factors of $a c$	Sum of Factors
$1$ and $12$	$1 + 12 = 13 ✓$
$2$ and $6$	$2 + 6 = 8 \times$
$3$ and $4$	$3 + 4 = 7 \times$

Factors of

a c

Sum of Factors

1

and

12

1 + 12 = 13 ✓

2

and

6

2 + 6 = 8 \times

3

and

4

3 + 4 = 7 \times

$a c$	$b$	Factors
Positive	Positive	Both positive
Positive	Negative	Both negative
Negative	Positive	One positive and one negative. The absolute value of the positive factor is greater.
Negative	Negative	One positive and one negative. The absolute value of the negative factor is greater.

a c

b

Factors

Positive

Both positive

Positive

Negative

Both negative

Negative

Positive

One positive and one negative. The absolute value of the positive factor is greater.

Negative

One positive and one negative. The absolute value of the negative factor is greater.

Factors of $a c = 6$	Sum of Factors
$- 1$ and $- 6$	$- 1 + (- 6) = - 7$
$- 2$ and $- 3$	$- 2 + (- 3) = - 5$

Factors of

a c = 6

Sum of Factors

- 1

and

- 6

- 1 + (- 6) = - 7

- 2

and

- 3

- 2 + (- 3) = - 5

Factors of $a c = - 56$	Sum of Factors
$- 56$ and $1$	$- 56 + 1 = - 55$
$- 28$ and $2$	$- 28 + 2 = - 26$
$- 14$ and $4$	$- 14 + 4 = - 10$
$- 8$ and $7$	$- 8 + 7 = - 1$

Factors of

a c = - 56

Sum of Factors

- 56

and

1

- 56 + 1 = - 55

- 28

and

2

- 28 + 2 = - 26

- 14

and

4

- 14 + 4 = - 10

- 8

and

7

- 8 + 7 = - 1

Factors of $a c = - 30$	Sum of Factors
$- 1$ and $30$	$29$
$- 2$ and $15$	$13$
$- 3$ and $10$	$7$
$- 5$ and $6$	$1$

Factors of

a c = - 30

Sum of Factors

- 1

and

30

29

- 2

and

15

13

- 3

and

10

7

- 5

and

6

1

Factors of $- 176$	Sum of Factors
$- 1$ and $176$	$- 1 + 176 = 175$
$- 2$ and $88$	$- 2 + 88 = 86$
$- 4$ and $44$	$- 4 + 44 = 40$
$- 8$ and $22$	$- 8 + 22 = 14$
$- 11$ and $16$	$- 11 + 16 = 5$
$- 16$ and $11$	$- 16 + 11 = - 5$
$- 22$ and $8$	$- 22 + 8 = - 14$
$- 44$ and $4$	$- 44 + 4 = - 40$
$- 88$ and $2$	$- 88 + 2 = - 86$
$- 176$ and $1$	$- 176 + 1 = - 175$

Factors of

- 176

Sum of Factors

- 1

and

176

- 1 + 176 = 175

- 2

and

88

- 2 + 88 = 86

- 4

and

44

- 4 + 44 = 40

- 8

and

22

- 8 + 22 = 14

- 11

and

16

- 11 + 16 = 5

- 16

and

11

- 16 + 11 = - 5

- 22

and

8

- 22 + 8 = - 14

- 44

and

4

- 44 + 4 = - 40

- 88

and

2

- 88 + 2 = - 86

- 176

and

1

- 176 + 1 = - 175

Factors of $6$	Sum of Factors
$1$ and $6$	$7$
$2$ and $3$	$5$

Factors of

6

Sum of Factors

1

and

6

7

2

and

3

5

Factor the following trinomial.

2 x^{2} + 2 x - 12

We will factor the given trinomial. To do so, we first need to check the greatest common factor (GCF) between the terms of the expression. To identify the GCF, we will split each term into its factors.

After factoring out the GCF, we obtained a trinomial with a leading coefficient of 1 in the parentheses. Now let's consider the coefficient of the linear term. x^2+x-6 ⇕ x^2+ 1x+( -6) Next, we will factor this expression into the form (x+ p)(x+ q), where p and q are factor pairs of -6 whose sum is 1. Let's list all factors of -6 and calculate their sums in a table.

Factors of -6	Sum of Factors
1 and -6	-5
2 and -3	-1
3 and -2	1
6 and -1	5

According to the table, 3 and -2 satisfy the conditions. Therefore, we will substitute p= 3 and q= -2 into the product (x+ p)(x+ q).

Finally, we can complete the factorization. Don't forget the 2 that we factored out at the beginning! 2x^2+2x-12 ⇕ 2(x+ 3)(x -2)

Factor the following trinomial.

6 m^{2} + 17 m + 5

We are given a quadratic trinomial in the form am^2+bm+c, where |a|≠1 and there are no common factors between its terms. To factor this expression, we will rewrite the middle term, bm, as a sum of two terms. The coefficients of these two terms will be factors of ac whose sum must be b. 6m^2+ 17m+ 5 In this case, a= 6, b= 17, and c= 5. To rewrite the expression, we need to follow these three steps.

First, find a c. Since a= 6 and c= 5, the value of a c is 6* 5=30.
Second, find the factors of a c. Since a c=30, which is positive, we need the factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Because b= 17 is also positive, those factors will need to be positive so that their sum is positive.

Factors of ac=30	Sum of Factors
1 and 30	31
2 and 15	17
3 and 10	13
5 and 6	11

Lastly, rewrite bm as the sum of two terms. Now that we know which factor pair to use, we can rewrite bm as the sum of two terms.

6m^2 + 17m+5 ⇕ 6m^2 + 2m + 15m+5 Now we can factor this expression by grouping.

Factor the following trinomial.

- 4 y^{2} - 8 y + 21

We will find the factored form of the given trinomial. To do so, we will begin by factoring out -1 from the quadratic expression to make its leading coefficient positive. -4y^2-8y+21 ⇔ - (4y^2+8y-21 ) We now have a quadratic trinomial in the form ay^2+by+c, where |a| ≠ 1 and the greatest common factor of the terms is 1. To factor this expression, we will rewrite the middle term, by, as the sum of two terms. The coefficients of these two terms will be factors of ac whose sum must be b. - ( 4y^2+8y-21 ) ⇔ - ( 4y^2+ 8y+( -21) ) In this expression, a= 4, b= 8, and c= -21. There are now three steps we need to follow to rewrite this expression.

First, find a c. Since we have that a= 4 and c= -21, the value of a c is 4( -21)=-84.
Second, find the factors of a c. Since 4( -21)=-84, which is negative, we need factors of a c to have the different signs — one positive and one negative — for the product to be negative. Since b= 8, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor.

Factors of ac=-84	Sum of Factors
-1 and 84	83
-2 and -42	40
-3 and 28	25
-4 and 21	17
-6 and 14	8
-7 and 12	5

Lastly, rewrite by as the sum of two terms. Now that we know which factor pair to use, we can rewrite by as the sum of two terms.

- (4y^2 + 8y-21 ) ⇕ - ( 4y^2 - 6y + 14y-21 ) Now, we can factor this obtained expression by grouping.

Solve the following equation by factoring.

2 k^{2} + k - 15 = 0

We want to solve the given equation by factoring. Let's write an outline of our plan.

Determine if anything can be factored from the original equation.
Find the product of the coefficients a and c.
Determine which factor pair can be used.
Write the the pair as the sum of two terms and determine what can be factored.
Apply the Zero Product Property.

To get things started, notice that the greatest common factor of the terms is 1. That means we do not need to factor anything out of the original equation. We are working with a trinomial. Let's then identify the equation's coefficients. 2k^2+k-15=0 ⇕ 2k^2+ 1k+( -15)=0 Here, a= 2, b= 1, and c= -15. Let's use this information find the value of a c. 2( -15)=-30 Now, let's find the factors of a c by analyzing -30, the product of 2 by -15. Because -30 is a negative value, we need the factors of a c to have different signs — one positive and one negative. a + times a - equals a - Recall that b= 1, which is positive. That means the absolute value of the positive factor will need to be greater than the absolute value of the negative factor.

Factors of ac=- 30	Sum of Factors
-1 and 30	- 1+30=29
-2 and 15	- 2+15=13
-3 and 10	- 3+10=7
-5 and 6	- 5+6=1

The factor pair of 6 and ¬5. We can now rewrite bk as the sum of two terms. 2k^2+ 1k-15=0 ⇕ 2k^2 - 5k + 6k-15=0 Next, we will factor the expression on the left-hand side of the equation by grouping.

Finally, we will apply the Zero Product Property to find the solutions to the given equation.

The solutions to the equation are k= 52 and k=-3.

Factoring Any Quadratic Expression

Catch-Up and Review

Finding a Pair of Numbers

Factoring a Quadratic Trinomial

Factoring the Face Area

Hint

Solution

Factoring a Quadratic Trinomial

Factoring $a x^{2} + b x + c$ When $a c$ is Positive

Hint

Solution

Factoring $a x^{2} + b x + c$ When $a c$ is Negative

Hint

Solution

Factoring $a x^{2} + b x + c$ When $a$ is Negative

Hint

Solution

The Width of a Path

Hint

Solution

Finding a Pair of Numbers

Hint

Solution

Factoring Any Quadratic Expression

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	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions