Know the Proof

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Know the Proof

• The Quadratic Formula

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Proof of the Quadratic Formula
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Proof of the Quadratic Formula
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Proof of the Quadratic Formula
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Proof of the Quadratic Formula
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The Discriminant

• If D gt 0 and D Perfect Square ?
• 2 Rational Real Roots
• If D gt 0 and D Non-Perfect Square ?
• 2 Irrational Real Roots
• If D 0 ? One Rational Root
• If D lt 0 ? Two Imaginary Roots

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Lesson 5 Contents
Example 1 Two Rational Roots Example 2 One
Rational Root Example 3 Irrational Roots Example
4 Complex Roots Example 5 Describe Roots
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Example 5-1a
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Example 5-1a
Then, substitute these values into the Quadratic
Formula.
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Example 5-1a
Answer The solutions are 11 and 3.
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Example 5-1b
Answer 2, 15
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Example 5-2a
Identify a, b, and c. Then, substitute these
values into the Quadratic Formula.
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Example 5-2a
Answer The solution is 17.
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Example 5-2b
Answer 11
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Example 5-3a
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Example 5-3a
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Example 5-3a
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Example 5-3b
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Example 5-4a
Now use the Quadratic Formula.
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Example 5-4a
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Example 5-4a
A graph of the function shows that the solutions
are complex, but it cannot help you find them.
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Example 5-4a
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Example 5-4b
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Example 5-5a
Answer The discriminant is 0, so there is one
rational root.
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Example 5-5a
Answer The discriminant is negative, so there
are two complex roots.
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Example 5-5a
Answer The discriminant is 80, which is not a
perfect square. Therefore, there are two
irrational roots.
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Example 5-5a
Answer The discriminant is 81, which is a
perfect square. Therefore, there are two
rational roots.
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Example 5-5b
Answer 0 1 rational root
Answer 24 2 complex roots
Answer 5 2 irrational roots
Answer 64 2 rational roots