Difference of Two Squares on the GMAT

- By Katharine Rudzitis

Many mathematical expressions on the GMAT can be factored using the difference of two squares formula: x2 – y2 = (x + y)(x – y). Applying this formula can save time, but sometimes it’s tough to recognize a difference of two squares in action. So,, here are two examples where we can apply the formula.

Recognizing Squares

Sometimes it’s easy to see when a polynomial can be factored using the difference of two squares. Other times, it may take a few steps to find the squares. Sometimes, rewriting a term as a squared number is a helpful first step.

Let’s factor the following expression: 100x2 – 1

At first this may not look like a factorable expression, but the trick is to recognize that 1 = 12. So, we can rewrite the expression as:

100x2 – 12

Now this looks like the difference of two squares. We know that (10x)(10x) = 100x2, so the entire expression becomes:

(10x)2 – 12

This matches the formula for the difference of two squares, so the expression can be simplified to

(10x + 1)(10x – 1).

Solving with the Difference of Two Squares

Some GMAT questions may not explicitly ask test-takers to factor a polynomial. Consider this problem:

Let x2 - y2 = 20 and x - y = 5. Solve for x and y.

At first, it might seem best to solve for x in terms of y and plug in that value to the first equation. Applying the difference of two squares method will make this problem much easier.

The first equation is already in the correct format for the difference of two squares, so we can rewrite it as:

(x + y)(x - y) = 20.

The problem also tells us that (x - y) = 5, which we can plug in:

(x + y)(5) = 20.

Dividing both sides by 5 gives us:

(x + y) = 4

Now we have two different equations: x + y = 4 and x – y = 5.

We can solve this system for x and y in a variety of ways. One approach is to add the two equations to get: 2x = 9, and then divide both sides by 2 to get x = 4.5

Now we can take the equation x + y = 4 and replace x with 4.5 to get: 4.5 + y = 4, which means y = -0.5

So, our solution is x = 4.5 and y = -0.5

We’ve solved for both x and y using the difference of two squares formula. On the GMAT, always be on the lookout for ways to apply this formula, because it can help you reach answers faster.


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