Algebra

1
Algebra

• 10.4 Solving Equations in Factored Form

2
Quadratic Equations in Standard Form

• You have already learned the standard form of a
  quadratic equation

ax² bx c 0 where a ? 0
3
Quadratic Equations in Factored Form
Consider the equation x² 8x 15 0
Here is the same equation, in factored form
(x 5)(x 3) 0
FOIL to be sure
x²
3x
5x
15
x² 8x 15 0
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Two Ways to Solve x² 8x 15 0
Quadratic Formula
Factored form
(x 5)(x 3) 0
v
-8 8² - 4(1)(15)
x
This means that for the equation to be true,
either
2(1)
v
-8
4
x
or
x 3 0
x 5 0
2
-3 -3
-5 -5
-8
2
-6
-10
x

x -3
x -5
2
2
2
x -3, -5
x -3, -5
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ZERO PRODUCT PROPERTY
If a product equals zero, then one of the factors
must be 0.
a 0
If ab 0, then either
b 0
or both 0
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Apply the ZERO PRODUCT PROPERTY
We said that the solutions to the equation x²
8x 15 0 or (x 5)(x 3) are -3 and
-5.
Substitute each solution into the equation in
factored form
(-3 5)(-3 3) 0
Substitute -3 for x

(2) (0) 0 TRUE
(-5 5)(-5 3) 0
Substitute -5 for x

(0) (-2) 0 TRUE
Substitute into original equation
(-3)² 8(-3) 15 0
True
(-5)² 8(-5) 15 0
True
7
Try these
x 7 0 or x 3 0
(x 7) (x 3) 0
x -7 or x 3
x - 4 0 or x 5 0
(x - 4)(x 5) 0
x 4 or x 5
2x 5 0 or x 3 0
(2x 5)(x 3) 0
5 5
3 3
2x 5
x 3
2 2
5
x
x 3
2
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Shortcut
When the factors are in the form (x __ ) or (x -
__ ) the solutions will always be the opposite of
the factors, in this case -7 and 3.
(x 7) (x 3) 0
Ask yourself what value of x will make this
binomial 0?
Ask yourself what value of x will make this
binomial 0?
-7
3
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Shortcut
(3x 2) (x 1) 0
Ask yourself what value of x will make this
binomial 0?
The other solution is 1
-2
(3 2)
0
3
The denominator would cancel with 3 when
multiplied
The numerator would cancel with 2 when added
2
One solution is
-
3
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Try these
x -½ or x 3
(2x 1) (x 3) 0
4
5
(3x - 4)(2x 5) 0
x or x
-
3
2
5
(3x 5)(2x 3) 0
3
x or x
-
3
2
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A Few together from the HW

• P. 600 29, 43

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Homework

• pg. 600 19 - 46, 69-77