With the power of factoring, you can solve most quadratic equations you will
encounter. Factoring involves breaking up the quadratic expression (usually the
left side of the equation) down from a trinomial—which has the form ax^2 + bx +
c —into a product of two binomials—which has the form (dx + f)(gx + h). To do
this, we’re going to use the coefficients of the trinomial (the a, b, and c
numbers), as well as the grouping method we covered in the previous section.
We’re also going to employ the wise, age-old, foolproof system of . . . trial and
error.
Here’s a sample quadratic equation: 2x^2 – x – 6 = 0. To the left of the equals
sign, we have our happy little trinomial expression (the “2x^2 – x – 6” part), and
we can recognize that it has the form ax^2 + bx + c. First, we are going to need to
extract the three coefficients a, b, and c. In this case, a = 2, b = –1, and c = –6.
The game we have to play now is to find two numbers that multiply up to the
quantity a × c, but also add up to the b in the trinomial. So in this case because a
× c = –12, we need two numbers that multiply to –12, but also add to –1. Let’s
try making a list of all the factor pairs of –12 and checking their sums.
Factoring is best mastered by practice. Go ask your old algebra teacher for some practice worksheets—you’ll gain insane math skillsAND crazy admiration from your prof.
—Samantha
So the important numbers we want to save from this whole experience are: –4
and 3. Here’s how we’ll use those numbers to factor. Let’s go back to our
trinomial:
2 x^2 – x – 6We’re going to take the middle term (the –x) and rewrite it as the sum of two
like terms, using the two numbers we just found (–4 and 3). Since that –x can be
represented by –4x + 3x, we can rewrite the original trinomial as a four term
expression: