5.2 Solving Quadratic Equations by Factoring Chapter 5 - Quadratic
5.2 Solving Quadratic Equations by Factoring
NOTES
Chapter 5 - Quadratic Functions
5.2 Solving Quadratic Equations by Factoring Goals: Factor quadratic expressions Solve quadratic equations Find zeros of quadratic functions (real-life examples)
2
Factor the trinomial in the form x + bx + c 2
2
Factor the trinomial in the form x + bx + c 2
x + 16x + 28
x + 5x - 14
c = 28 --> list the factors of 28
c = -14 --> list the factors of -14
b = 16 --> which factors listed above will combine to (positive) 16?
b = 5 --> which factors listed above will combine to 5?
* check your answer by using the FOIL method
(x + __)(x - __)
Now, you try it! Factor the trinomial. 2
1. x + 5x + 4
* check your answer by using the FOIL method
(x + __)(x - __)
Now, you try it! Factor the trinomial. 2
4. x + 6x = 27
2
2. x + 8x + 12 2
5. x = -2x + 80
2
3. x - 4x + 3 2
6. x - 72 = 14x
Chapter 5 - Quadratic Functions
1
5.2 Solving Quadratic Equations by Factoring
The Box Method for Factoring a Polynomial in the form ax2 + bx + c EXAMPLE:10x2 + 11x − 6
NOTES The Box Method for Factoring a Polynomial EXAMPLE:10x2 + 11x − 6 Step 1 - create a 2x2 box
Step 1 - create a 2x2 box Step 2 - in the top left corner put the first term and in the bottom right corner put the last term. Step 3 - multiply these two terms together to get −60x2 . Find two factors of −60x2 that when added together they will give you the middle term 11x. These are 15x and −4x. Put these into the open boxes. Step 4 - factor the terms in each row and in each column. Step 5 - the sum of the factors for the columns and the sum of the factors for the rows are the polynomial’s factors: (2x + 3)(5x − 2)
The Box Method for Factoring a Polynomial
The Box Method for Factoring a Polynomial
EXAMPLE:10x2 + 11x − 6
2 EXAMPLE:10x + 11x − 6 1
Step 2 - in the top left corner put the first term and in the bottom right corner put the last term.
Step 3 - multiply these two terms together to get −60x2 . Find two factors of −60x2 that when added together they will give you the middle term 11x. These are 15x and −4x. Put these into the open boxes.
1 60
10x2
10x2
15x
2 30 3 20
-6
-4x
4 15
-6
5 12 6 10
The Box Method for Factoring a Polynomial
The Box Method for Factoring a Polynomial
1EXAMPLE:10x2 + 11x − 6
1EXAMPLE:10x2 + 11x − 6 Step 5 - the sum of the factors for the columns and the sum of the factors for
Step 4 - factor the terms in each row and in each column.
2x
3
5x
10x2
15x
-2
-4x
-6
Chapter 5 - Quadratic Functions
the rows are the polynomial’s factors:
2x
3
5x
10x2
15x
-2
-4x
-6
(2x + 3)(5x − 2)
2
5.2 Solving Quadratic Equations by Factoring
NOTES
Now, YOU TRY IT!
Now, YOU TRY IT!
1. Factor the trinomial 3x2 − 17x + 10 (using the box method)
2. Factor the trinomial 8x2 + 18x + 9 (using the box method)
Now, you try it! Factor the following trinomials.
Factor out a GCF 1st! Always check to see whether the terms have a common
2
3. 5x - 7x + 2
2
monomial factor.
example: 5x + 15x - 20
2
1. 4x + 28x + 40
2
2. 3x + 24x + 36
2
4. 2x + 7x + 3
2
3. 2x - 8x + 6 2
5. 7x - 4x - 3
Factoring with Special Patterns
Factoring with Special Patterns
Recall: Difference of Two Squares
Recall: Perfect Square Trinomial
pattern 2
2
a - b = (a + b)(a - b)
example 2
x - 9 = (x + 3)(x - 3)
Now, you try it! Factor the following expressions 2
1. x - 36 2
2. x - 81 2
3. 4x - 25 2
4. 16x - 9
Chapter 5 - Quadratic Functions
pattern
example
2
2
2
2
2
2
a + 2ab + b = (a + b) a - 2ab + b = (a - b)
2
x + 12x + 36 = (x + 6) 2
x - 8x + 16 = (x - 4)
2
2
Now, you try it! Factor the following expressions 2 1. x + 4x + 4 2
2. x - 6x + 9 2
3. 9x + 24x + 16 2
4. 49x - 14x + 1
3
5.2 Solving Quadratic Equations by Factoring Remember... Factoring a Polynomial Step 1: Standard Form Step 2: Factor out the GCF
NOTES
Solve the Quadratic Equation 2x(x - 7) = 0
Step 3: Factor (special product, box method, ...) Step 4: Check to see that it is completely factored 2
4x + 12x = 0 Next... Solve a Quadratic by Factoring Step 1: Standard Form Step 2: Factor Step 3: Zero Product Property (all factors set = to zero)
2
x + 7x 18 = 0
Step 4: Solve each equation Step 5: Check answer
Solve the Quadratic Equation 2
3x + 10 = x
Now, you try it! Solve the Quadratic Equation 1. (x + 1)(x + 4) = 0
2
4. x + 6x = 27
2
x = 18x 81
2
5. 3x = -6x + 240
2
6. 2x - 144 = 28x
2. x + 8x + 12 = 0
2
2
2x + 12x = 14
Now, you try it! Solve the Quadratic Equation 2
7. 5x - 7x + 2 = 0
3. x - 4x + 3 = 0
2
Now, you try it! Solve the Quadratic Equation 2
10. x - 36
=0
2
11. 4x - 25 = 0 2
8. 2x + 7x + 3 = 0 2
12. x + 4x + 4 = 0
2
9. 7x - 4x - 3 = 0
Chapter 5 - Quadratic Functions
2
13. x - 6x + 9 = 0
4
NOTES
Chapter 5 - Quadratic Functions
5.2 Solving Quadratic Equations by Factoring Goals: Factor quadratic expressions Solve quadratic equations Find zeros of quadratic functions (real-life examples)
2
Factor the trinomial in the form x + bx + c 2
2
Factor the trinomial in the form x + bx + c 2
x + 16x + 28
x + 5x - 14
c = 28 --> list the factors of 28
c = -14 --> list the factors of -14
b = 16 --> which factors listed above will combine to (positive) 16?
b = 5 --> which factors listed above will combine to 5?
* check your answer by using the FOIL method
(x + __)(x - __)
Now, you try it! Factor the trinomial. 2
1. x + 5x + 4
* check your answer by using the FOIL method
(x + __)(x - __)
Now, you try it! Factor the trinomial. 2
4. x + 6x = 27
2
2. x + 8x + 12 2
5. x = -2x + 80
2
3. x - 4x + 3 2
6. x - 72 = 14x
Chapter 5 - Quadratic Functions
1
5.2 Solving Quadratic Equations by Factoring
The Box Method for Factoring a Polynomial in the form ax2 + bx + c EXAMPLE:10x2 + 11x − 6
NOTES The Box Method for Factoring a Polynomial EXAMPLE:10x2 + 11x − 6 Step 1 - create a 2x2 box
Step 1 - create a 2x2 box Step 2 - in the top left corner put the first term and in the bottom right corner put the last term. Step 3 - multiply these two terms together to get −60x2 . Find two factors of −60x2 that when added together they will give you the middle term 11x. These are 15x and −4x. Put these into the open boxes. Step 4 - factor the terms in each row and in each column. Step 5 - the sum of the factors for the columns and the sum of the factors for the rows are the polynomial’s factors: (2x + 3)(5x − 2)
The Box Method for Factoring a Polynomial
The Box Method for Factoring a Polynomial
EXAMPLE:10x2 + 11x − 6
2 EXAMPLE:10x + 11x − 6 1
Step 2 - in the top left corner put the first term and in the bottom right corner put the last term.
Step 3 - multiply these two terms together to get −60x2 . Find two factors of −60x2 that when added together they will give you the middle term 11x. These are 15x and −4x. Put these into the open boxes.
1 60
10x2
10x2
15x
2 30 3 20
-6
-4x
4 15
-6
5 12 6 10
The Box Method for Factoring a Polynomial
The Box Method for Factoring a Polynomial
1EXAMPLE:10x2 + 11x − 6
1EXAMPLE:10x2 + 11x − 6 Step 5 - the sum of the factors for the columns and the sum of the factors for
Step 4 - factor the terms in each row and in each column.
2x
3
5x
10x2
15x
-2
-4x
-6
Chapter 5 - Quadratic Functions
the rows are the polynomial’s factors:
2x
3
5x
10x2
15x
-2
-4x
-6
(2x + 3)(5x − 2)
2
5.2 Solving Quadratic Equations by Factoring
NOTES
Now, YOU TRY IT!
Now, YOU TRY IT!
1. Factor the trinomial 3x2 − 17x + 10 (using the box method)
2. Factor the trinomial 8x2 + 18x + 9 (using the box method)
Now, you try it! Factor the following trinomials.
Factor out a GCF 1st! Always check to see whether the terms have a common
2
3. 5x - 7x + 2
2
monomial factor.
example: 5x + 15x - 20
2
1. 4x + 28x + 40
2
2. 3x + 24x + 36
2
4. 2x + 7x + 3
2
3. 2x - 8x + 6 2
5. 7x - 4x - 3
Factoring with Special Patterns
Factoring with Special Patterns
Recall: Difference of Two Squares
Recall: Perfect Square Trinomial
pattern 2
2
a - b = (a + b)(a - b)
example 2
x - 9 = (x + 3)(x - 3)
Now, you try it! Factor the following expressions 2
1. x - 36 2
2. x - 81 2
3. 4x - 25 2
4. 16x - 9
Chapter 5 - Quadratic Functions
pattern
example
2
2
2
2
2
2
a + 2ab + b = (a + b) a - 2ab + b = (a - b)
2
x + 12x + 36 = (x + 6) 2
x - 8x + 16 = (x - 4)
2
2
Now, you try it! Factor the following expressions 2 1. x + 4x + 4 2
2. x - 6x + 9 2
3. 9x + 24x + 16 2
4. 49x - 14x + 1
3
5.2 Solving Quadratic Equations by Factoring Remember... Factoring a Polynomial Step 1: Standard Form Step 2: Factor out the GCF
NOTES
Solve the Quadratic Equation 2x(x - 7) = 0
Step 3: Factor (special product, box method, ...) Step 4: Check to see that it is completely factored 2
4x + 12x = 0 Next... Solve a Quadratic by Factoring Step 1: Standard Form Step 2: Factor Step 3: Zero Product Property (all factors set = to zero)
2
x + 7x 18 = 0
Step 4: Solve each equation Step 5: Check answer
Solve the Quadratic Equation 2
3x + 10 = x
Now, you try it! Solve the Quadratic Equation 1. (x + 1)(x + 4) = 0
2
4. x + 6x = 27
2
x = 18x 81
2
5. 3x = -6x + 240
2
6. 2x - 144 = 28x
2. x + 8x + 12 = 0
2
2
2x + 12x = 14
Now, you try it! Solve the Quadratic Equation 2
7. 5x - 7x + 2 = 0
3. x - 4x + 3 = 0
2
Now, you try it! Solve the Quadratic Equation 2
10. x - 36
=0
2
11. 4x - 25 = 0 2
8. 2x + 7x + 3 = 0 2
12. x + 4x + 4 = 0
2
9. 7x - 4x - 3 = 0
Chapter 5 - Quadratic Functions
2
13. x - 6x + 9 = 0
4
