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Obj. 3 Quadratic Equations
Unit 1 Functions and Relations
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Concepts and Objectives
Quadratic Equations
Simplify complex numbers Solve quadratic equations, finding all solutions
Cubic Equations
Solve the sum or difference of two cubes
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Complex Numbers
As you should recall, there is no real number solution to
x2 = 1so the number i has been defined so that
i2 = 1
which means that . Complex numbers arenumbers in the form a + bi, where a and b are real
numbers.
For any positive real number a,
= 1i
=a i a
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Complex Numbers
Example: Simplify (a) (b) (c)
(a)
(b)
(c)
16 48 ( )( ) 4 9
= =16 16 4i i
= =48 48 4 3i i
( )( ) = =4 9 36 6
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Quadratic Equations
A quadratic equation is an equation that can be written
in the form
where a, b, and c are real numbers, with a 0. This is
standard form.
A quadratic equation can be solved by factoring,
graphing, completing the square, or by using the
quadratic formula.
Graphing and factoring dont always work, butcompleting the square and the quadratic formula will
always provide the solution(s).
+ + =2
0ax bx c
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Factoring Quadratic Equations
Factoring works because of thezero-factor property:
Ifa and b are complex numbers with ab = 0, then a =0 or b = 0 or both.
To solve a quadratic equation by factoring:
Put the equation into standard form (=0). Find two numbers which multiply to ac and add to b.
Split the b term using these two numbers.
Find the GCF of each pair.
Use the distributive property to re-write the equation
into the factors.
Set each factor equal to zero and solve.
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Factoring Quadratic Equations
Example: Solve by factoring.
The solution set is
=2
2 15 0x x
= = = 2, 1, 15a b c 30
1
6 5 + =22 6 5 15 0x x x
( ) ( ) + =2 3 5 3 0x x x
( )( )+ =2 5 3 0x x
+ = =2 5 0 or 3 0x x
= 5
, 32x
5
,32
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Quadratic Formula
The solutions of the quadratic equation ,
where a 0, are
Example: Solve
+ + =2
0ax bx c
=
24
2
b b acx
a
= 2
2 4x x
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Quadratic Formula
Example: Solve
The solution set is
= 2
2 4x x
+ =22 4 0x x = = =2, 1, 4a b c
( ) ( ) ( )( )
( )
=
2
1 1 4 2 4
2 2x
= =
1 1 32 1 31
4 4
=
1 31
4
i
1 31
4 4i
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Cubic Equations
A cubic equation is an equation of degree 3. We will
mainly be working with cubic equations that are the sumor difference of two cubes:
a3 b3 = 0
Equations of this form factor as
To solve this, set each factor equal to zero and solve.
(Use the Quadratic Formula for the quadratic factor.)
( )( ) + =2 2 0a b a ab b
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Cubic Equations
Example: Solve 8x3 + 125 = 0 ( ) + =3 3
2 5 0x
( )( )+ + =22 5 4 10 25 0x x x
+ = + =2
2 5 0 or 4 10 25 0x x x
+ =
=
=
2 5 0
2 5
5
2
x
x
x
( ) ( )( )( )
=
2
10 10 4 4 252 4
x
= = =
10 300 10 10 3 5 5 3
8 8 4
i i
5 5 5 3,
2 4 4
iThe solution set is
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Homework & Practice Problems
Page 109: 20-40 (5s)
Page 119: 15-30 (5s), 45-55 (5s), 62, 64