Warm Up

1
Warm Up

• Solve each equation by graphing
• Approximate solution(s) to the nearest thousandth.

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Chapter 11 Section 1

• Quadratic Equations

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The General Form of a Quadratic Equation
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Recall How We Solved Them in Chapter 6
Factoring
Graphical
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What are the possible number of solutions to a
quadratic equation?
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Solving Quadratic Equations

• Recall that a Quadratic Equation is another name
  for a polynomial equation of degree 2.
• We have seen that some (but not all) quadratic
  equations can be solved by factoring.
• We have seen that solving by graphing is a
  solution method that works for every quadratic
  equation, but it is only an approximate method
  (in general).
• Today we will see two new algebraic methods for
  finding exact solutions to quadratic equations.

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Principle of Square Roots

• For any real number k, if x2 k, then

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The Principle of Square Roots

• The Principle of Square Roots
• If , then
• Note A convenient shorthand is
• Examples
• If then
• If then
• If then

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Solve the Following Algebraically and Graphically
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Your Turn

• Lets revisit the warm up. You solved these three
  quadratic equations (approximately) by graphing.
• Now solve them algebraically, using the principle
  of square roots. How do your exact (algebraic)
  solutions compare to your graphical (approximate)
  solutions?

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Chapter 11 Section 2

• The Quadratic Formula

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x
1
1
x
x
x
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x
x
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x
x
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x
x
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Completing the Square

• Complete the square on the following

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The Quadratic Formula

• Our previous method, using the principle of
  square roots, works well if an equation can be
  reduced to the form (where x stands for either a
  variable or an expression), but not all quadratic
  equations can be reduced to this form.
• The Quadratic Formula is an exact method for
  solving any quadratic equation. It is not always
  the fastest method, but it is guaranteed to work!

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The Quadratic Formula

• Recall that the standard form of a quadratic
  equation is , where .
• Definition The Quadratic Formula
• For a quadratic equation in standard form, the
  solutions are given by
• Notice that the notation allows us to
  represent two solutions using a single equation.

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The Quadratic FormulaExamples

• Solve

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The Quadratic FormulaYour Turn

• Solve

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General Strategy forSolving Quadratic Equations

• If the quadratic equation can be reduced to the
  form , use the principle of square roots.
• Otherwise, put the quadratic equation into
  standard form, then
• Try to solve by factoring.
• If factoring seems difficult or impossible, use
  the quadratic formula.

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Side Trip

• Complex NumbersDefinition
• A complex number is any number that can be
  written in the form

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Example