Squaring numbers near 50 and 100 and the ones ending with 5 TRICKS

You can often use the difference of squares pattern, a^2 - b^2 = (a+b)(a-b), to calculate squares quickly. We just need to subtract a square so that at least one of the factors (a+b) or (a-b) will be easy to work with. So if we wanted to compute the value of 85^2, we could set up an equation where we subtract 5^2 on the left side, use the difference of squares factorization on the right side, then solve for 85^2:

\(\begin{align}\\
85^2 - 5^2 &= (85 + 5)(85-5) \\\\
85^2 - 25 &= (90)(80) \\\\
85^2 &= 7200 + 25 = 7225\\
\end{align}\)

We choose to subtract 5^2 above because then the difference of squares factorization will give us two simple numbers to multiply. Subtracting 15^2 would also work out cleanly. Similarly, if we want to calculate the value of 94^2, we can subtract 6^2. Then the (a+b) factor ends up being 100, which is easy to multiply by:

\(\begin{align}\\
94^2 - 6^2 &= (94+6)(94-6) \\\\
94^2 - 36 &= (100(88) \\\\
94^2 &= 8800 + 36 = 8836\\
\end{align}\)

One advantage to using the difference of squares here is that you don't need to learn different techniques for different types of numbers. We can use it for some three-digit numbers as well, so if we want to compute 997^2, we can subtract 3^2 first:

\(\begin{align}\\
997^2 - 3^2 &= (997 + 3)(997 - 3) \\\\
997^2 - 9 &= (1000)(994) \\\\
997^2 &= 994000 + 9 = 994009 \\
\end{align}\)

But there's a more important advantage for GMAT test takers. While you almost never need to calculate awkward squares on the GMAT, you almost always need to use the difference of squares somewhere. And that's the main reason I like the approach above - even if a test taker never needs to use it to actually compute a square, practicing it may help him or her to recognize opportunities to use the difference of squares pattern in the kinds of questions that do appear on the GMAT.