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Binomials and Quadratics
A binomial is an expression with two terms.
These terms are usually a number and a variable. If you stick your
hand in a jar of binomials and grab a bunch, here’s what you might
have in the palm of your hand:
On the SAT, most binomials take on the form of a variable
(with or without a coefficient) and a number.
When two binomials get together, they really like to party.
And by party, we mean “multiply all terms together.”
It’s not our idea of a good time, but let’s not judge binomials.
Multiplying two binomials is like a double dose of distribution.
You can use the acronym FOIL—shorthand for first, outer, inner, last—to
remember how to do it.
- F Multiply the two first terms of each binomial together.
- O Multiply the two outside terms together.
- I Multiply the two inside terms together.
- L Multiply the two last terms of each binomial together.
The expression you get after multiplying using FOIL is 
.
If you make this puppy equal to zero, you get a quadratic equation.
Because the new SAT places more emphasis on binomials and quadratics,
expect to see at least a couple of them.
.
If you make this puppy equal to zero, you get a quadratic equation.
Because the new SAT places more emphasis on binomials and quadratics,
expect to see at least a couple of them. The problem is you can’t do that much with 
.
To figure out what x might be, you have to deduce
what two binomials combine together to make 
.
The kicker is that there’s no set, easy way to do this. Where’s
FOIL when you need it? Well, it’s around, kind of. Think of FOIL
as a casual friend helping you move. FOIL will move the lamps and
silverware, but it’s not going to do any heavy lifting.
.
To figure out what x might be, you have to deduce
what two binomials combine together to make .
The kicker is that there’s no set, easy way to do this. Where’s
FOIL when you need it? Well, it’s around, kind of. Think of FOIL
as a casual friend helping you move. FOIL will move the lamps and
silverware, but it’s not going to do any heavy lifting. Look at the first term in the equation 
.
Because you know from FOIL that the y2 is
made by multiplying the first two terms together, you can figure
out that the first term of each binomial must be y,
because when you multiply y with y,
you get y2. Like a puzzle, here’s where
you are now:
.
Because you know from FOIL that the y2 is
made by multiplying the first two terms together, you can figure
out that the first term of each binomial must be y,
because when you multiply y with y,
you get y2. Like a puzzle, here’s where
you are now:
You figure out the next two terms in tandem. Look at the
last term, 20, and come up with all the different sets of numbers
that can be multiplied together to make 20. Don’t forget the negatives,
either.
| Factors of 20 | |
| 1 | 20 |
| –1 | –20 |
| 2 | 10 |
| –2 | –10 |
| 4 | 5 |
| –4 | –5 |
Look at the middle term on this chart, –9y.
Which set of factors, if you added them together, would equal –9?
The answer’s there at the end, –4 and –5. If these are the last
terms, then their product is 20 and their sum is –9y.
You have cracked the quadratic and now know that:
So what is the value of y? Well, the
whole equation equals zero, so one of those binomials must also
equal zero because anything multiplied by zero is zero. There’s
no way to tell which one is the culprit, so the answer is y =
4, 5.
As you can see, quadratic equations require a good deal
of number grinding, and your calculator can’t really help you with
it. That’s why they appear on the SAT.
The Quadratic Formula
This formula is like a first-aid kit from World War I.
It should be used only if all else fails. Our previous quadratic
equation factored very nicely into the variable y and
the integers –4 and –5. If you come across a quadratic equation
that doesn’t factor neatly, you can still pull out an answer using
the quadratic formula.
Quadratic equations take the form ax2+bx+c=0,
where a does not equal zero. The quadratic formula
states:
This doesn’t exactly roll off the tongue. In our nice
quadratic 
, a = 1, b =
–9, and c = 20. Plugging these values in would
give us:
, a = 1, b =
–9, and c = 20. Plugging these values in would
give us:
As you can see, we get the same answer, showing that the
quadratic formula works. As you can also see, factoring is a much
cleaner process than working through the quadratic formula.
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