6.6 Solving Quadratic Equations

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6.6 Solving Quadratic Equations

• Objectives
• Multiply binominals using the FOIL method.
• Factor Trinomials.
• Solve quadratic equations by factoring.
• Solve quadratic equations using the quadratic
  formula.
• Page 317

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• A binomial expression has just two terms (usually
  an x term and a constant). There is no equal
  sign.  Its general form is ax b, where a and b
  are real numbers and a ? 0.
• One way to multiply two binomials is to use the
  FOIL method. FOIL stands for the pairs of terms
  that are multiplied First, Outside, Inside,
  Last.
• This method works best when the two binomials
  are in standard form (by descending exponent,
  ending with the constant term).
• The resulting expression usually has four terms
  before it is simplified. Quite often, the two
  middle (from the Outside and Inside) terms can be
  combined.

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For example
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• The opposite of multiplying two binomials is to
  factor or break down a polynomial (many termed)
  expression.
• Several methods for factoring are given in the
  text. Be persistent in factoring! It is normal to
  try several pairs of factors, looking for the
  right ones.
• The more you work with factoring, the easier it
  will be to find the correct factors.
• Also, if you check your work by using the FOIL
  method, it is virtually impossible to get a
  factoring problem wrong. 
• Remember!  When factoring, always take out any
  factor that is common to all the terms first.

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• A quadratic equation involves a single variable
  with exponents no higher than 2.
• Its general form is where a,
  b, and c are real numbers and . 
• For a quadratic equation it is possible to have
  two unique solutions, two repeated solutions (the
  same number twice), or no real solutions.
• The solutions may be rational or irrational
  numbers.

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• To solve a quadratic equation, if it is
  factorable
•     1.  Make sure the equation is in the general
  form. 
•     2.  Factor the equation.
•     3.  Set each factor to zero. 
•     4.  Solve each simple linear equation.

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To solve a quadratic equation if you cant factor
the equation

• Make sure the equation is in the general
  form. 
• Identify a, b, and c.
• Substitute a, b, and c into the quadratic
  formula
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• Simplify.
•  

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• The beauty of the quadratic formula is that it
  works on any quadratic equation when put in the
  form general form. 
• If you are having trouble factoring a problem,
  the quadratic formula might be quicker.
• Always be sure and check your solution in the
  original quadratic equation.

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Find the product
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Factor x2 - 7x 12.

• 1. Pairs of numbers which make 12 when
  multiplied (1, 12), (2, 6), and (3, 4).
• 1 12?7. 2 6?7. 3 4 7. Thus, d 3 and
  e 4.
• (x - 3)(x - 4)
• Check (x - 3)(x - 4) x2 -4x - 3x 12 x2 -
  7x 12
• Thus, x2 - 7x 12 (x - 3)(x - 4).

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Factor 2x3 4x2 2x.

• First, remove common factors 2x3 4x2 2x
  2x(x2 2x 1)
• Pairs of numbers which make 1 when multiplied
  (1, 1).
• 1 1 2. Thus, d 1 and e 1.
• 2x(x 1)(x 1) (don't forget the common
  factor!)
• Check 2x(x 1)(x 1) 2x(x2 2x 1) 2x3
  4x2 2x
• Thus, 2x3 4x2 2x 2x(x 1)(x 1) 2x(x
  1)2.x2 2x 1 is a perfect square trinomial.

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The Box Method for Factoring a Polynomial
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The Box Method for Factoring a Polynomial
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Factor the trinomial
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Use the Quadratic Formula to solve
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Solve for x
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Solve for x
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Solve using the quadratic formula
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Homework Assignment on the Internet

• Section 6.6 (Read Solving Quadratic Equation)
• Pp 329-330 2-78even.