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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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6.6 Solving Quadratic Equations

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6 6 solving quadratic equations l.jpg

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 can’t 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.


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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.


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