6'2 Solving Quadratic Equations by Graphing 13107

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6.2 Solving Quadratic Equations by Graphing
1/31/07

• Quadratic Equation can be written in the form
  ax2 bx c 0, where a ? 0.
• The solutions of a quadratic equation are called
  the roots.
• One method for finding the roots of a quadratic
  equation is to find the zeroes of the related
  quadratic functions.
• The zeroes of the function are the x-intercepts
  of its graph.
• These are the solutions of the related equation
  because f(x) 0 at those points.

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Two Real Solutions

• Solve x2 6x 8 0 by graphing.
• 1. Graph the related quadratic function.
• f(x) x2 6x 8
• 2. The equation of the axis of symmetry is x
  - 6 -3
• 
  2(1)
• 3. Make a table using x values around -3.
  Then, graph each point. From the table and the
  graph, you can see the zeros of the function are
  -4 and -2. Therefore, the solutions of the
  equation are -4 and -2.

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Solutions of a Quadratic Equation

• A quadratic equation can have one real solution,
  two real solutions, or no real solution.
• One Real solution Two Real Solutions
• No Real Solution

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One Real Solution

• Solve 8x x2 16
• Write the equation in ax2 bx c 0 form
• 8x x2 16 ? -x2 8x 16 0
• Graph f(x) -x2 8x 16 0
• The graph has only one x-intercept, 4. So the
  solution is 4.

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No Real Solution

• Find two real numbers whose sum is 6 and whose
  product is 10 or show that no such numbers exist.
• Let x one of the numbers. Then 6 x the
  other number
• Since the product of the two numbers is 10, you
  know that x (6 x) 10
• x ( 6 x ) 10
• 6x x2 10
• -x2 6x 10 0
• Graph f(x) -x2 6x 10 0.
• The graph doesnt have any x-intercepts, so the
  original equation has no real solution. It is
  not possible for two numbers to have a sum of 6
  and a product of 10.

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Estimate Roots

• Often exact roots cannot be found by graphing.
  So, in this case, you can estimate solutions by
  stating the consecutive integers between which
  the roots are located.
• Solve x2 4x 1 0 by graphing. If exact
  roots cant be found, stat the consecutive
  integers between which the roots are located
• Axis of symmetry x - 4 2
• 2(-1)
• The x-intercepts of the graph are between 0 and 1
  and between 3 and 4. So, one solution is between
  0 and 1, and the other is between 3 and 4.

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More Practice!!!

• Textbook p. 297 4 12 even
• Homework Worksheet 6.2