5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
5.3 FACTORING QUADRATICS
There are several methods available for
solving a quadratic equation:
1.
2.
3.
4.
5.
By Square Roots
By Factoring
By Completing the Square
By the Quadratic Formula
By Graphing
The method depends on the form of the equation.
FACTORING QUADRATIC TRINOMIALS
Example: 5x2 + 17x + 14
1. The expression must be in ascending or
descending order.
2. Make a sum/product chart.
3. Divide each number by the leading coefficient.
4. Reduce each fraction if possible.
5. Denominator = constant or coefficient of first term
Numerator = constant or coefficient of last term
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
Examples:
a. x2 + 6x + 8
b. 3x2 - 11x + 6
Examples:
c. x2 + 7x - 18
d. 3x2 +10x - 8
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
Practice
Factor each trinomial.
1) x2 - 16x + 39
2) x2 + 2x - 35
3) x2 + 22x + 121
4) x2 - 2x - 63
5) 14x2 - 11x + 2
6) 12x2 + 16x - 3
7) 2x2 + 13x + 6
8) 9x2 - 9x - 28
Answers
Factor each trinomial.
1) x2 - 16x + 39
(x - 3)(x - 13)
2) x2 + 2x - 35
(x + 7)(x - 5)
3) x2 + 22x + 121
(x + 11)(x + 11)
4) x2 - 2x - 63
(x + 7)(x - 9)
5) 14x2 - 11x + 2
(7x - 2)(2x - 1)
6) 12x2 + 16x - 3
(2x + 3)(6x - 1)
7) 2x2 + 13x + 6
(2x + 1)(x + 6)
8) 9x2 - 9x - 28
(3x + 4)(3x - 7)
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
Special Factoring Patterns
1. FACTORING DIFFERENCE OF SQUARES
x2 ­ 4 = (x ­ 2)(x + 2)
4x2 ­ 9 = (2x ­ 3)(2x + 3)
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x ­ 49 = (x ­ 7)(x + 7)
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64x2 ­ 25 = (8x ­ 5)(8x + 5)
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a2 ­ b2 =
Special Factoring Patterns
2. PERFECT SQUARE TRINOMIALS
x2 + 14x + 49 = (x + 7)2
x2 ­ 8x + 16 = (x ­ 4)2
4x2 ­ 20x + 25 = (2x ­ 5)2
9x2 + 12x + 4 = (3x + 2)2
a2 ­ 2ab + b2 =
a2 + 2ab + b2 =
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5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
Practice
Factor completely.
1. 4x2 - 121
2. 9x2 - 24x + 16
3. 225 - x2
4. x2 + 10x + 25
5. 10x2 - 13x - 3
Answers
Factor completely.
1. 4x2 - 121
2. 9x2 - 24x + 16
(2x - 11)(2x + 11)
(3x - 4)2
3. 225 - x2
(15 - x)(15 + x)
5. 10x2 - 13x - 3
(2x - 3)(5x + 1)
4. x2 + 10x + 25
(x + 5)2
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
When factoring,
ALWAYS look for the GCF first!
Greatest Common Factor
the largest factor that divides ALL of the terms
a. 12x2 - 3
b. 7v2 - 42v
FACTOR COMPLETELY
d. 15x2 + 6x
c. 5x2 - 45
e. 3x2 - 9x + 6
f. 36x - 48x2 + 24x3
5.3 Factoring and Solving Quadratics (work).notebook
Practice
Factor completely.
1. 12x2 - 3
2. 45x 2 + 10x
3. 8x2 - 24x + 18
4. x 2 + 5x + 4
5. 6x2 + 13x - 5
Answers
Factor completely.
1. 12x2 - 3
2. 45x 2 + 10x
5x(9x + 2)
3(2x - 1)(2x + 1)
3. 8x2 - 24x + 18
2(2x - 3) 2
5. 6x2 + 13x - 5
(2x + 5)(3x - 1)
4. x 2 + 5x + 4
(x + 1)(x + 4)
October 21, 2016
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
FACTORING FOUR TERMS
When factoring four terms, use the
grouping method.
a. x2 - 12x + 3x - 36
b. ra + rb + sa + sb
FACTOR USING THE GROUPING METHOD.
c. y2 - 12y - 4y + 48 d. k2 + 3k - 8k - 24
5.3 Factoring and Solving Quadratics (work).notebook
Practice
Factor completely.
1. 2x2y - x + 6xy - 3
2. 6cd2 - 8cd - 9d + 12
3. 2xz - 6xy + 2yz - 6y2
Answers
Factor completely.
1. 2x2y - x + 6xy - 3
(2xy - 1)(x + 3)
2. 6cd2 - 8cd - 9d + 12
(2cd - 3)(3d - 4)
3. 2xz - 6xy + 2yz - 6y2
2(x + y)(z - 3y)
October 21, 2016
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
STEPS TO FACTOR POLYNOMIALS
Step 1: Factor out any GCF.
Step 2: For a binomial with the x-term missing,
check if it is the difference of two squares.
Step 3: For a trinomial, check to see if it
matches the perfect square trinomial
pattern or jump into a sum/product chart.
Step 4: For 4 terms, use grouping.
Step 5: See if any factors can be factored further.
Solving
ax2 + bx + c = 0
by
FACTORING
5.3 Factoring and Solving Quadratics (work).notebook
October 21, 2016
Quadratic Equations In Standard Form
ax2 +bx + c = 0
NOTE:
The solutions of a quadratic equation
are called the roots of the equation.
AND
Since the function's value (y) is zero
when ax2 + bx + c = 0, the solutions
are also called zeros of the function
f(x) = ax2 +bx + c.
To solve ax2 +bx + c = 0:
Use the "zero product property".
If A B = 0, then A = 0 or B = 0
a. 3x - 6 = x2 - 10
1. Set = to 0 (may need to move terms).
2. Factor.
3. Set each factor = to 0.
4. Solve for the variable.
5.3 Factoring and Solving Quadratics (work).notebook
b. Find the zeros of f(x) = 3x2 + 10x - 8.
c. What are the roots of the equation
x2 - 5x - 36 = 0?
October 21, 2016
5.3 Factoring and Solving Quadratics (work).notebook
d. 3x2 + 4x = 4
f. 3x2 +24x + 45 = 0
October 21, 2016
e. 16x2 = 49
g. 10x2 = 9x
5.3 Factoring and Solving Quadratics (work).notebook
Practice
Solve by factoring.
1. 4x2 = 24x
2. 16x2 - 361 = 0
3. 20x = 25x2 + 4
4. 2x2 + 7x - 15 = 0
Answers
Solve by factoring.
1. 4x2 = 24x
x = 0, 6
2. 16x2 - 361 = 0
x = + 19/4
3. 20x = 25x2 + 4
x = 2/5
4. 2x2 + 7x - 15 = 0
x = -5, 3/2
October 21, 2016
5.3 Factoring and Solving Quadratics (work).notebook
Word Problems
AGAIN!!
Doubling Area
Extra
Example
You have a rectangular vegetable garden in your backyard
that measures 15 feet by 10 feet. You want to double the
area of the garden by adding the same distance x to the
length and width of the garden. Find the value of x and
the new dimensions of the garden.
October 21, 2016