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SAT Difference of Squares Shortcut

A difference of squares is any expression of the form something squared minus something else squared, and it factors instantly into two matching binomials. The SAT plants this pattern in simplification questions, in rational expressions, and in questions with large numbers where recognizing it saves a minute of arithmetic.

The rule

a squared minus b squared always factors into the quantity a plus b times the quantity a minus b.

The pattern is that a squared minus b squared equals the quantity a plus b times the quantity a minus b. So x squared minus 49 factors into the quantity x plus 7 times the quantity x minus 7. Checking by expanding shows why the middle terms vanish: you get x squared minus 7x plus 7x minus 49, and the two middle terms cancel exactly.

The pieces do not have to be single variables. The expression 9x squared minus 25 is the quantity 3x squared minus the quantity 5 squared, so it factors into the quantity 3x plus 5 times the quantity 3x minus 5. And x to the fourth minus 16 factors first into the quantity x squared plus 4 times the quantity x squared minus 4, and that second factor is itself a difference of squares, giving the quantity x plus 2 times the quantity x minus 2.

A sum of squares does not factor this way. There is no real factorization of x squared plus 49 into binomials with real numbers, so if you see a plus between two squares, the pattern does not apply. The SAT includes a choice built from the wrong assumption that it does.

Keep the difference of squares separate from the perfect square patterns. The quantity a plus b squared equals a squared plus 2ab plus b squared, and the quantity a minus b squared equals a squared minus 2ab plus b squared. Those have a middle term. The difference of squares is the case where the middle term cancels, and that only happens when the two binomials have opposite signs.

The pattern also speeds up plain arithmetic. To compute 51 squared minus 49 squared, do not square anything. Write it as the quantity 51 plus 49 times the quantity 51 minus 49, which is 100 times 2, or 200. When a question shows two large squares being subtracted, it is usually pointing at this shortcut.

The trap

Applying the pattern to a sum of squares such as x squared plus 16, which does not factor over the real numbers.

Try it

Try one
Real question

The graphs of y = 2x and y = x^2 intersect at two points. What is the nonzero x-coordinate of an intersection point?

The graphs of y = x² - 4x and y = x + k are shown in the xy-plane, where k is a constant. The graphs intersect at (6, 12) and at one other point. What is the value of k?

  • A) 0
  • B) 2
  • C) 6 ✓
  • D) 12

Why: The point (6, 12) lies on the line y = x + k, so substitute it: 12 = 6 + k. Subtracting 6 from both sides gives k = 6, which is also the height where the line crosses the y-axis. "12" is the y-coordinate you were handed, used without subtracting the x-coordinate. "2" comes from dividing 12 by 6 instead of subtracting 6 from 12. "0" would make the line y = x, which passes through (6, 6) rather than (6, 12).

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Questions students ask

How do I recognize a difference of squares quickly?

Check three things: exactly two terms, a subtraction between them, and both terms perfect squares. Coefficients count, so 25x squared qualifies because 25 is 5 squared and x squared is a square. If all three hold, the factorization is immediate.

What is the difference between this and a perfect square trinomial?

A perfect square trinomial has three terms and comes from squaring a single binomial, so it keeps a middle term of twice the product. The difference of squares has only two terms, because the middle terms cancel when the two binomials differ by a sign.

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