Binomials and Quadratic Equations
With its new emphasis on Algebra II, the new SAT asks
quite a few more questions on binomials. What is a binomial? Quite
simply, it is an expression that has two terms:
are both binomials.
The multiplication of binomials is its own SAT topic and
a fundamental skill for dealing with the dreaded quadratic equations.
Luckily, the best acronym ever made (other than SCUBA) will help
you remember how to multiply binomials. This acronym is FOIL,
and it stands for First, Outer + Inner, Last.
The acronym tells you the order in which you multiply the terms
of two binomials to get the correct product.
For example, let’s say you were kidnapped by wretched
fork-tongued lizard-men whose only weakness was binomials. Now what
if the lizard-king asked to you to multiply the binomials:
(x + 1)(x +
What would you do? Follow FOIL, of course. First, multiply
the first terms of each binomial:
Next, multiply the outer terms of the binomials:
Then, multiply the inner terms:
And multiply the last terms:
Add all these terms together:
Finally, combine like terms, and you get
Here are a few more examples of multiplied binomials to
use to test yourself.
Quadratics are the robots who return from the future to
destroy humankind at the end of Terminator 3
the future is now: The new SAT forces you to take on quadratic equations.
But the future also isn’t that tough. Here’s the first thing: A
quadratic expression takes the form ax2 + bx + c
. And here’s the second: Note how
closely ax2 + bx + c
the products formed when binomials are multiplied. Coincidence?
A quadratic equation sets a quadratic equal to zero: ax2 + bx + c =
0. The values of x for which
the equation holds are called the roots, or solutions, of the quadratic
equation. Some of the SAT questions on quadratic equations ask you
to find their roots.
There are two basic ways to find roots: by factoring and
by using the quadratic formula. Factoring is faster, but doesn’t
always work. The quadratic formula takes longer to work out, but
works on every quadratic equation.
On the SAT, you’ll be able to factor almost every
quadratic expression or equation that appears, but every once in
a while the test may throw in a quadratic that you need to know
the quadratic formula to solve. In other words, you probably don’t
need to know the quadratic formula, and if you’re pressed for time
you could survive if you didn’t spend time studying it. But if you
want to slam dunk the new SAT, memorize it.
Here’s why quadratic expressions resemble the product
of two binomials: Quadratic expressions are the
product of two binomials. Factoring a quadratic means breaking the
quadratic back into its binomial parts. Factoring might as well
be called LIOF; it’s FOIL in reverse. Check out this quadratic expression:
For starters, you know that the binomials have the form (x + m)(x + n),
where m and n are
constants. How do you know this? Because of that x2.
When you FOIL to get the first term, you multiply the two first
terms of the binomials. To get x2,
you have to multiply x by x.
As for figuring out what m and n are,
you have two clues to work with.
The sum of m and n is 10,
since the 10x is derived from multiplying
the OUTER and INNER terms of the binomials and then adding the resulting
terms together (10x = mx + nx,
so m + n must equal 10).
product of m and n equals 21,
since 21 is the product of the two last terms of the
binomials. The only pair of numbers that fit the bill for m and n are 3 and 7 (3
+ 7 = 10 and ), so x2 +
10x + 21 = (x + 3)(x +
But what if this had been a quadratic equation rather
than a plain old quadratic expression? Well, first of all, it would
have looked like this: x2 +
10x + 21 = 0. Second, once you’d factored
it to get (x + 3)(x + 7)
= 0, you could solve for its roots. Because the product of
two terms is zero, one of the terms must be equal to zero. Since x +
3 = 0 or x + 7 = 0, the solutions
(also known as the roots) of the quadratic must be x =
–3 and x = –7.
Quadratics with Negative Terms
Once you get the hang of it, factoring a quadratic with
negative terms is no harder than dealing with one with only positive
Consider the quadratic equation x2 –
4x – 21 = 0. Here’s what we know about this
equation: The first term of each binomial is x,
since the first term of the quadratic is x2;
the product of m and n is –21;
and the sum of a and b equals –4.
The equation also tells you that either m or n must
be negative, but that both cannot be negative,
because only the multiplication of one positive and one negative
number can result in a negative number. Now you need to look for
the numbers that fit these requirements for m and n.
The numbers that multiply together to give you –21 are –21 and 1, –7 and 3, –3 and 7,
and 21 and –1. Of these, the pair that
sums to –4 is –7 and 3. The
factoring of the equation is (x – 7)(x +3)
= 0. So the roots of the equation are x =
7 and x = –3.
Two Special Quadratics
There are two special quadratics that pop up all the time
on the SAT. If you know what they look like and can identify them
quickly, you’ll save time. These two quadratics are called the “perfect
square” and the “difference of two squares.”
Perfect square quadratics are the product of a term squared
(multiplied by itself). There are therefore two kinds of perfect
square quadratics: those formed by the squaring of a binomial of
the form (a + b)2 and
those formed by the squaring of binomials that look like (a – b)2.
a2 + 2ab + b2 =
(a + b)(a + b)
= (a + b)2
Example: a2 +
6ab + 9 = (a + 3)2
a2 – 2ab + b2 =
(a – b)(a – b)
= (a – b)2
Example: a2 –
6ab + 9 = (a –3)2
Note that when you solve for the roots of a perfect square
quadratic equation, the solution for the equation (a + b)2 =
0 will be –b, while the solution
for (a + b)2 =
0 will be b.
The difference of two-squares quadratic equations follow
the form below:
See how the middle term drops out? The disappearance of
the middle term causes lots of students to fail to recognize when
they’re dealing with a difference of two-squares quadratic.
Since the ability to factor quadratics relies in large
part on your ability to “read” the information in the quadratic,
the best way to sharpen your eye is to practice, practice, practice.
Take a look at the following examples and try to factor them on
your own before you peek at the answers.