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Section 8.3 The Discriminant
and the Nature of Solutions
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The Discriminant
Type and Number of Solutions
Writing Equations from Solutions
8.3
1
Introducing … The Discriminant!
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is the Radicand Part of the Quadratic Equation
 b  b 2  4ac
x
2a
It predicts the types of solutions.
If b2 – 4ac is
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positive:
two different real numbers
0:
one real (two equal real numbers)
negative:
two different complex numbers
positive perfect square: two different rational numbers
positive but imperfect: two
different irrational numbers
8.3
2
What Types of Solutions?
b2 – 4ac
9 x  12 x  4  0
2
(12) 2  4(9)( 4)  144  144
x  5x  8  0
2
(5) 2  4(1)(8)  25  32
2x  7x  3  0
2
(7) 2  4(2)( 3)  49  24
x 40
2
(0)  4(1)( 4)  0  16
2
0
7
73
16
8.3
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Writing Equations from Solutions
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We can use the reverse of the Principle of Zero Products
(x – 2)(x + 3) = 0 means solutions x = 2 and x= -3
Think: x2 + x – 6 = 0 is equivalent to 2x2 + 2x – 12 = 0
Many quadratic equations can have the same solutions
5 )0
(
x

3
)(
x

2
Find an equation having solutions:
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x = 3 and x = 5/2
( x  2i )( x  2i )  0
x = ±2i
x  4i  0
x 40
x = ±5 7
x = 0, x = -4 and x = 1
x( x  4)( x  1)  0
2
2
x( x 2  3x  4)  0
8.3
2
x 2  112 x  15 2  0
2 x 2  11x  15  0
( x  5 7 )( x  5 7 )  0
x 2  25  7  0
x 2  175  0
x 3  3x 2  4 x  0
4
What Next? Quadratic Applications
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Section 8.4
8.3
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