Download 2.3 Quadratic Factoring

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ACT
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F-IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.*
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A-CED.1: Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential functions.*
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Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
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A-REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y =
g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find successive approximations. Include
cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.*
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F-IF.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima.
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F-IF.8a: Use the process of factoring and completing the square in a quadratic function to show zeros,
extreme values, and symmetry of the graph, and interpret these in terms of a context.
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A-REI.4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.
Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real
numbers a and b.
Bell Ringer
* Is a value of the input X that makes the output f(x) equal zero.
* The zeros of a function are the x-intercepts
What are the zeros of a function ?
* Zero-Product Property:
* If ab = 0, then a = 0 or b = 0; Example: If (x + 3)(x-7) = 0, then (x + 3) = 0 or (x - 7) = 0.
First sign
Second sign
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(
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+
+
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+
-
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+
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* Solve by Factoring
Product of
ac
x +6x = -8
2
2
x + 6x + 8 = 0
Add
(S)
Sum of
a+b
Multiply
(P)
(x + 2)(x + 4)
FOIL
set = to 0 ?
a=1?
P
S
1 8
2 4
9
6
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2
x + 3x = -2
2
x + 3x + 2 = 0
zeros
(x + 1)(x + 2)
set = to 0 ?
a=1?
P S
1 2
3
x2 + 8x + 7
x2 + 6x + 4
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x2 + 12x + 32
Factor x2 – 17x + 72
Reminder: find factors with product ac and a + b.
(
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- )(
- )
x2 – 7x + 12
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x2 – 11x + 24
x2 – 14x - 32
x2 + 3x – 10
( +
)( - )
x2 – 11x = - 15
* Write in standard form
* Factor
* Use the zero-product property
* Solve for x
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1)
2)
3)
Factor x2 – 5x - 6
Factor x2 + 7x + 10
Factor x2 – 3x + 2
Pick up Factoring worksheet; BEGIN! We work bell-to-bell
Worksheet due next class
Bell Ringer
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1)
x2 – 5x - 6
2)
x2 + 5x + 6
3)
x2 – 5x + 6
Factor and apply Zero-Product Property
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2
x  27  6x
x  9  0 or x  3  0
x  9 or x  3
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x  32x 1  0
x  3  0 or 2x  1  0
x  3 or 2x  1
1
x  3 or x 
2
The solutions are -3 and 1/2.
*Factoring : Perfect Squares
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In this example, you
must first factor the
equation. Notice the
familiar pattern.
2
9x  4  0
Factor using “difference of two squares.”
3x  23x  2   0
3x  2  0 or 3x  2  0
3x  2 or 3x  2
2
2
x   or x 
3
3
2
81x - 36
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2
16x - 6561
2
4x + 25
* Compare these two functions:
use a Venn Diagram
2
X –9=0
X–9=0
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