6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 377 + + =⋅+⋅+⋅+⋅ abcd acadbcbd ( )( ) Multiply Binomials Before discussing quadratic equations, we begin by discussing binomials and multiplying binomials. A binomial is an expression that contains two terms in which each exponent that appears on a variable is a whole number. + + − − x x x x Examples of binomials 3 3 5 5 4 2 To multiply two binomials, we use the distributive property, which was discussed in Section 5.5. Each term in the first binomial must be multiplied by each term in the second binomial. A convenient way to carry out this process is the FOIL method . The acronym FOIL indicates that you multiply the F irst terms, O uter terms, I nner terms, and L ast terms of the binomials, in that order. I F L O First Outer Inner Last After multiplying the first, outer, inner, and last terms, combine all like terms. Example 1 Multiplying Binomials Multiply x x ( 4)( 7). + + Solution The FOIL method of multiplication yields + + =⋅+⋅+⋅+⋅ = + + + = + + x x x x x x x x x x x ( 4)( 7) 7 4 4 7 7 4 28 11 28 2 2 F O I L Note that the like terms x7 and x4 were combined to get x 11 . 7 Now try Exercise 7 Example 2 Multiplying Binomials Multiply x x (3 4)(2 7). − + Solution − + = ⋅ + ⋅ + − ⋅ + − ⋅ = + − − = + − x x x x x x x x x x x (3 4)(2 7) 3 2 3 7 (4)2 (4)7 6 21 8 28 6 13 28 2 2 F O I L 7 Now try Exercise 11 Timely Tip When we use FOIL to multiply two binomials, we are really using the distributive property twice. + + = + + + = + + + a b c d a c d b c d ac ad bc bd ( )( ) ( ) ( ) Factoring Trinomials of the Form x bx c 2 + + The expression x x8 15 2 + + is an example of a trinomial. A trinomial is an expression containing three terms in which each exponent that appears on a variable is a whole number. Example 1 showed that x x x x ( 4)( 7) 11 28 2 + + = + + Since the product of x 4 + and x 7 + is x x 11 28, 2 + + we say that x 4 + and x 7 + are factors of x x 11 28. 2 + + To factor an expression means to write the
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