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GRE Algebra

There are common algebraic patterns that the new GRE wants you to recognize. Before we discuss these, I need to mention the FOIL method.

The FOIL method is one that almost everybody remembers learning at some point circa middle school. Though you may have forgotten the details, with a little practice (and you definitely want to become adept at the foil method), you should be able to use it effectively.

First off , FOIL stands for First, Outer, Inner, and Last, and refers to the position of numbers and/or variables within parenthesis. Let’s have a look:

(x - y)*(x + y) = ?

Remember, parenthesis stand for multiplication. The tricky part is how to multiply together a bunch of x’s and y’s. The answer: the FOIL method.

  • F (First): The first term in each parentheses is x, so we multiply the x’s together to get: x^2
  • O (Outer): The term on the outside of the left parenthesis is ‘x’ and on the outside of the right parenthesis is y. We multiply the two together to get: xy.
  • I (Inner): Now we multiply the inner terms in each parenthesis: ({-y})(x) = -xy
  • L (Last): Finally, we multiply the terms that are the rightmost to get ({-y})(y) = -y^2

Now we add together our results F + O + I + L: x^2 - xy + xy - y^2 = x^2 - y^2


So (x - y) (x + y) = x^2 - y^2.

Memorize this. Do not spend time on the test actually completing the steps above.


Other important algebraic expressions to memorize are:

(x-y)(x -y) = x^2 - 2xy + y^2

(x + y)(x + y) = x^2 + 2xy + y^2.


Here are some examples in which we apply the above.

  1. (x - 5) (x + 5) = x^2 - 25
  2. (a + 3)(a + 3) = a^2 + 6a + 9
  3. (b - 2c)(b - 2c) = b^2 - 4bc + 4c^2


Reversing FOIL and Solving for Roots of Equation

In other instances, you will take an equation in which you have to turn into parentheses.


Other applications of FOIL

(98)(102) = ?

(79)(81) = ?


These questions appear as though they would not relate to the FOIL method. But upon closer inspection, we can see that these numbers aren’t random.

If we add them, instead of multiplying, we get 200, for the first question, and 160, for the second. Or 100 + 100 and 80 + 80.

Let’s focus on the pair of hundreds: 100 * 100 vs. 98 * 102.

Notice that (98)(102) can be written as (100 - 2)(100 + 2). Now you should see the (x - y)(x + y) form, which expressed correctly is 100^2 - 2^2 = 9,996.

Solving in this way is much more effective, because 100^2 = 10,000, a number you should know of the top of your head.


Other Tips

Compare the following:

(x - 2)^2 = 4 vs. 2 = sqrt{x - 2}

In the first case, there are two solutions: 0 and 4. Remember when you square a negative number, as in (-2)^2 = 4, you get a positive number.

With the equation on the right side, the one in which x is under the square root sign, if you get a negative number, you cannot take the square root of it (at least in GRE world, where imaginary numbers do not come into play).

In the case of 2 = sqrt{x-2}, we square both sides to get 4 = x - 2, so x = 6.


Blog posts about Algebra