5.3 Factoring and Solving Quadratics...
Transcript of 5.3 Factoring and Solving Quadratics...
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
The method depends on the form of the equation.
There are several methods available for solving a quadratic equation:
1. By Square Roots2. By Factoring3. By Completing the Square4. By the Quadratic Formula5. By Graphing
5.3 FACTORING QUADRATICS
FACTORING QUADRATIC TRINOMIALS
2. Make a sum/product chart.
5x2 + 17x + 14Example:
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
1. The expression must be in ascending or descending order.
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
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
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
(x - 3)(x - 13) (x + 7)(x - 5)
(x + 11)(x + 11) (x + 7)(x - 9)
(7x - 2)(2x - 1) (2x + 3)(6x - 1)
(2x + 1)(x + 6) (3x + 4)(3x - 7)
Answers
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
Special Factoring Patterns
FACTORING DIFFERENCE OF SQUARES
x2 4 = (x 2)(x + 2)
4x2 9 = (2x 3)(2x + 3)
x2 49 = (x 7)(x + 7)
64x2 25 = (8x 5)(8x + 5)
a2 b2 =
1.
What is the pattern?
Special Factoring Patterns
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 =
2.
What is the pattern?
a2 + 2ab + b2 =
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
3. 225 - x2 4. x2 + 10x + 25
5. 10x2 - 13x - 3
(2x - 11)(2x + 11) (3x - 4)2
(15 - x)(15 + x) (x + 5)2
(2x - 3)(5x + 1)
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 COMPLETELYc. 5x2 - 45 d. 15x2 + 6x
e. 3x2 - 9x + 6 f. 36x - 48x2 + 24x3
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
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
3. 8x2 - 24x + 18 4. x 2 + 5x + 4
5. 6x2 + 13x - 5
3(2x - 1)(2x + 1) 5x(9x + 2)
2(2x - 3)2 (x + 1)(x + 4)
(2x + 5)(3x - 1)
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
When factoring four terms, use the grouping method.
FACTORING FOUR TERMS
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 October 21, 2016
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
2. 6cd2 - 8cd - 9d + 12
3. 2xz - 6xy + 2yz - 6y2
(2xy - 1)(x + 3)
2(x + y)(z - 3y)
(2cd - 3)(3d - 4)
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
byFACTORING
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
The solutions of a quadratic equation are called the roots of the equation.
Quadratic Equations In Standard Formax2 +bx + c = 0
ANDSince 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.
NOTE:
Use the "zero product property".
To solve ax2 +bx + c = 0:
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 October 21, 2016
b. Find the zeros of f(x) = 3x2 + 10x - 8.
c. What are the roots of the equationx2 - 5x - 36 = 0?
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
d. 3x2 + 4x = 4 e. 16x2 = 49
f. 3x2 +24x + 45 = 0 g. 10x2 = 9x
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
PracticeSolve by factoring.
1. 4x2 = 24x
2. 16x2 - 361 = 0
3. 20x = 25x2 + 4
4. 2x2 + 7x - 15 = 0
AnswersSolve by factoring.
1. 4x2 = 24x
2. 16x2 - 361 = 0
3. 20x = 25x2 + 4
4. 2x2 + 7x - 15 = 0
x = 0, 6
x = + 19/4
x = 2/5
x = -5, 3/2
5.3 Factoring and Solving Quadratics (work).notebook October 21, 2016
Word Problems AGAIN!!
Doubling Area
Extra ExampleYou 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.