Solving Quadratic Equation by Graphing and Factoring

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Solving Quadratic Equation by Graphing and
Factoring

• Section 6.2 6.3
• CCSS A.REI.4b

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Mathematical Practices

• 1. Make sense of problems and persevere in
  solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the
  reasoning of others.  
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for and express regularity in repeated
  reasoning. 

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CCSS A.REI.4b

• SOLVE quadratic equations by inspection (e.g.,
  for x2 49), taking square roots, completing the
  square, the quadratic formula and factoring, as
  appropriate to the initial form of the equation.
  RECOGNIZE when the quadratic formula gives
  complex solutions and write them as a bi for
  real numbers a and b.

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Essential Question

• How do I determine the domain, range, maximum,
  minimum, roots, and y-intercept of a quadratic
  function from its graph how do I solve
  quadratic functions by factoring?

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Quadratic Equation

• y ax2 bx c
• ax2__ is the quadratic term.
• bx--- is the linear term.
• c-- is the constant term.
• The highest exponent is two therefore, the
  degree is two.

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Identifying Terms

• Example f(x)5x2-7x1
• Quadratic term 5x2
• Linear term -7x
• Constant term 1

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Identifying Terms

• Example f(x) 4x2 - 3
• Quadratic term 4x2
• Linear term 0
• Constant term -3

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Identifying Terms

• Now you try this problem.
• f(x) 5x2 - 2x 3
• quadratic term
• linear term
• constant term

5x2 -2x 3
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Quadratic Solutions

• The number of real solutions is at most two.

No solutions
One solution
Two solutions
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Solving Equations

• When we talk about solving these equations, we
  want to find the value of x when y 0. These
  values, where the graph crosses the x-axis, are
  called the x-intercepts.
• These values are also referred to as solutions,
  zeros, or roots.

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Identifying Solutions

• Example f(x) x2 - 4

Solutions are -2 and 2.
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Identifying Solutions

• Now you try this problem.
• f(x) 2x - x2
• Solutions are 0 and 2.

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Graphing Quadratic Equations

• The graph of a quadratic equation is a parabola.
• The roots or zeros are the x-intercepts.
• The vertex is the maximum or minimum point.
• All parabolas have an axis of symmetry.

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Graphing Quadratic Equations

• One method of graphing uses a table with
  arbitrary
• x-values.
• Graph y x2 - 4x
• Roots 0 and 4 , Vertex (2, -4) ,
• Axis of Symmetry x 2

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Graphing Quadratic Equations

• Try this problem y x2 - 2x - 8.
• Roots
• Vertex
• Axis of Symmetry

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Graphing Quadratic Equations

• The graphing calculator is also a helpful tool
  for graphing quadratic equations.

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Roots or Zeros of the Quadratic Equation

• The Roots or Zeros of the Quadratic Equation are
  the points where the graph hits the x axis.
• The zeros of the functions are the input that
  make the equation equal zero.
• Roots are 4,-3

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To solve a Quadratic Equation

• Make one side zero.
• Then factor then set each factor to zero

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Solve

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Solve

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Solve

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Solve

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Solve

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Solve
Solve

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Solve

• Multiply the ends together and find what adds to
  the coefficient of the middle term

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Solve

• Use -6 and 1 to break up the middle term

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Solve

• Use group factoring to factor, first two terms
  and then the last two terms

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Solve
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How to write a quadratic equation with roots

• Given r1,r2 the equation is (x - r1)(x - r2)0
• Then foil the factors,
• x2 - (r1 r2)x(r1 r2)0

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How to write a quadratic equation with roots

• Given r1,r2 the equation is (x - r1)(x - r2)0
• Then foil the factors,
• x2 - (r1 r2)x(r1 r2)0
• Roots are -2, 5
• Equation x2 - (-25)x(-2)(5)0
• x2 - 3x -10 0

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How to write a quadratic equation with roots

• Roots are ¼, 8
• Equation x2 -(¼8)x(¼)(8)0
• x2 -(33/4)x 2 0
• Must get rid of the fraction, multiply by the
  common dominator. 4
• 4x2 - 33x 8 0