Mathematics Notes by Success Tutorials: Factoring Quadratic Formulas

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1. Decomposition

mx² + nx + p      i.e.  4x² + 4x + 1

We will decompose this quadratic using the reverse of the FOIL method, as shown below.

The FOIL Method Explained

(a + b)(c + d) = ac + ad + bc + bd

                          F     O      I     L                 FOIL:    stands for First, Outer, Inner, Last.

(x + 1)(x + 2) = x² + 2x + 1x + 2

                      = x² + 3x + 2

FOIL    Note:  "ac" produces the first term "x^2"

                        "ad + bc" produces the second term "3x"

                        "bd" produces the third term "2"

Using the FOIL Method

mx² + nx + p = (a + b)(c + d)       i.e. 4x² + 4x + 1 = (a + b)(c + d)

mx² = ac                4x² = ac

nx = ad + bc          4x = ad + bc

p = bd                    1 = bd

What could "ac" be?

Possibilities are:  a = 2x, c = 2x, ac = 4x²

or

                          a = 1x, c = 4x, ac = 4x²

      First:     Try a = 2x, c = 2x

Now we have: (2x      )(2x      )

What could "bd" be?

Possibilities are: b = 1, d = 1, bd = 1

or b = -1, d = -1, bd = 1

Try b = 1, d = 1    because there are no negatives in this quadratic.

Now we have:   (2x + 1)(2x + 1) =  4x² + 2x + 2x + 1 =  4x² + 4x + 1

It works! This is the basic method known as decomposition to factor a quadratic.

2. Algebra Tiles

Algebra tiles have also been developed for factoring quadratics but they are far more cumbersome than decomposition and will not be discussed here. They involve physically moving small blocks around which represent parts of the quadratic.

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Opdateret d. 23/2/01