Section 11.3
Studying Solutions of
Quadratic Equations
Idea of 11.3
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Section 11.2 showed us how to find the answers.
Section 11.3 we show us what type of answers we will
have.
The types of answers are
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Repeated rational number
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Two distinct rational numbers.
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Two irrational conjugates
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Two complex conjugates
We will determine the type of answer by looking at the
determinant. The determinant is b² – 4ac.
Repeated Rational Number
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The answers will be a
repeated rational number
if b² - 4ac = 0
Example
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Determine the types of answers given
4x² – 12x + 9 = 0
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Therefore a = 4 b = -12 c = 9
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b² – 4ac
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(-12)² – 4(4)(9) = 144 – 144 = 0
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Therefore this equation will have one rational
number that will be repeated.
Two distinct rational numbers.
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The answers will be two
distinct rational numbers if
b² - 4ac is positive and a
perfect square.
Example
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Determine the types of answers given
4m² + 7m = 0
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Therefore a = 4 b = 7 c = 0
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b² – 4ac
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(7)² – 4(4)(0) = 49 – 0 = 49
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49 is positive and a perfect square, therefore
this equation will have two distinct rational
numbers.
Two Irrational Conjugates
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The answers will be two
irrational conjugate
numbers if b² - 4ac is
positive but not a perfect
square
Example
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Determine the types of answers given
10t² - t - 2 = 0
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Therefore a = 10 b = -1 c = -2
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b² – 4ac
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(-1)² – 4(10)(-1) = 1 + 40 = 41
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41 is positive, but is not a perfect square,
therefore this equation will have two distinct
irrational conjugates.
Two Complex Conjugates
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The answers will be two
complex conjugate
numbers if b² - 4ac is
negative
Example
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Determine the types of answers given
3x² + 5 = -7x
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Therefore a = 3 b = 7 c = 5
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b² – 4ac
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(7)² – 4(3)(5) = 49 – 60 = -11
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-11 is a negative number, therefore this
equation will have two complex conjugates.
How to find the equation
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How to find the equation given the solutions.
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Start with the answers equal to the variable.
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Write them as factors.
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Combine the factors through multiplication.
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FOIL
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Simplify
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Equation
Graphs and Discriminants
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The discriminant can help you create the graph.
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How?
Example
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Find the equation given the answers -6 and 3
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x = -6 or x = 3
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x + 6 = 0 or x – 3 = 0
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(x + 6)(x – 3) = 0*0
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X² – 3x + 6x - 18 = 0
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X² + 3x - 18 = 0
Example
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Find the equation given the answers 3, -1, 0
Homework
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Section 11.3
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# 7, 9, 13, 17, 29, 35, 53, 45, 56