Solving Quadratic Equations Section 1.3

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Solving Quadratic EquationsSection 1.3
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What is a Quadratic Equation?

• A quadratic equation in x is an equation that can
  be written in the standard form
• ax² bx c 0
• Where a,b,and c are real numbers and
• a ? 0.

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Solving a Quadratic Equation by Factoring.

• The factoring method applies the zero product
  property
• Words If a product is zero, then at least one
  of its
• factors has to be zero.
• Math If (B)(C)0, then B0 or C0 or both.

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Recap of steps for how to solve by Factoring

• Set equal to 0
• Factor
• Set each factor equal to 0 (keep the squared term
  positive)
• Solve each equation (be careful when determining
  solutions, some may be imaginary numbers)

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Example 1Solve x² - 12x 35 0 by factoring.

• Factor
• Set each factor equal to zero by the zero product
  property.
• Solve each equation to find solutions.
• The solution set is

• (x 7)(x - 5) 0
• (x 7)0 (x 5)0
• x 7 or x 5
• 5, 7

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Example 2Solve 3t² 10t 6 -2 by factoring.

• Check equation to make sure it is in standard
  form before solving. Is it?
• It is not, so set equation equal to zero first
• 3t² 10t 8 0
• Now factor and solve.
• (3t 4)(t 2) 0
• 3t 4 0 t 2 0
• t t -2

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Solve by factoring.
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Solve by the Square Root Method.

• If the quadratic has the form ax² c 0, where
  a ? 0, then we could use the square root method
  to solve.
• Words If an expression squared is equal to a
  constant, then that expression is equal to the
  positive or negative square root of the constant.
• Math If x² c, then x c.
• Note The variable squared must be isolated
  first (coefficient equal to 1).

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Example 1Solve by the Square Root Method

• 2x² - 32 0
• 2x² 32
• x² 16
• x 4

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Example 2Solve by the Square Root Method.

• 5x² 10 0
• 5x² -10
• x² -2
• x
• x i

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Example 3Solve by the Square Root Method.

• (x 3)² 25
• x 3 5
• x 3 5 or x 3 -5
• x 8 x
  -2

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Solve by the Square Root Method
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Solve by Completing the Square.

• Words
• Express the quadratic equation in the following
  form.
• Divide b by2 and square the result, then add the
  square to both sides.
• Write the left side of the equation as a perfect
  square.
• Solve by using the square root method.

• Math
• x² bx c
• x² bx ( )² c ( )²
• (x )² c ( )²

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Example 1Solve by Completing the Square.

• x² 8x 3 0
• x² 8x 3
• x² 8x (4)² 3 (4)²
• x² 8x 16 3 16
• (x 4)² 19
• x 4
• x -4

• Add three to both sides.
• Add ( )² which is (4)² to both sides.
• Write the left side as a perfect square and
  simplify the right side.
• Apply the square root method to solve.
• Subtract 4 from both sides to get your two
  solutions.

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Example 2Solve by Completing the Square when
the Leading Coefficient is not equal to 1.

• 2x² - 4x 3 0
• x² - 2x 0
• x² - 2x ___ ____
• x² - 2x 1 1
• (x 1)²
• x 1
• x 1

• Divide by the leading coefficient.
• Continue to solve using the completing the square
  method.
• Simplify radical.

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Quadratic Formula
If a quadratic cant be factored, you must use
the quadratic formula.

• If ax² bx c 0, then the solution is

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Solve
a 1 b -4 c -1
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Solve
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Solve
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Discriminant

• The term inside the radical b² - 4ac is called
  the discriminant.
• The discriminant gives important information
  about the corresponding solutions or answers of
  ax² bx c 0, where a,b, and c are real
  numbers.

b² - 4ac Solutions
b² - 4ac gt 0
b² - 4ac 0
b² - 4ac lt 0
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Tell what kind of solution to expect