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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 ... – PowerPoint PPT presentation

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Title: 6.6 Solving Quadratic Equations


1
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

2
  • 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.

3
For example
4
  • 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.

5
  • 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.

6
  • 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.

7
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
  •  
  • Simplify.
  •  

8
  • 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.

9
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10
Find the product
11
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).

12
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.

13
The Box Method for Factoring a Polynomial
14
The Box Method for Factoring a Polynomial
15
Factor the trinomial
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19
Use the Quadratic Formula to solve
20
Solve for x
21
Solve for x
22
Solve using the quadratic formula
23
Homework Assignment on the Internet
  • Section 6.6 (Read Solving Quadratic Equation)
  • Pp 329-330 2-78even.
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