Therefore, if you are aiming to tackle even the 160-170 questions on the Q section, you need to have some crafty mental math tricks up your sleeve. One of the most powerful involves the clever use of a famous algebra formula: the difference of two squares formula. See this page — "Quant: Difference of Two Squares" for uses of that formula in general problem solving. Here we will focus on factoring.
In general, factoring a big number can be time-consuming without a calculator. The QUANT might ask a question in which you need to know the factors, or the prime factorization, of a large number. See this page — "Math: Factors" — for more on prime factorizations.
First of all, notice how easy it is to square the multiples of
Those should all be recognizable as nice round perfect squares.
Now, suppose you are in a situation in which you have to factor, say,
In general, the QUANT is not going to put you in a situation in which you have to find the prime factorization of a general four digit number. If this situation does arise, you can bet there’s an enormously simplifying trick available, and factoring via the difference of two squares is an awfully likely candidate for that trick.
Just as the difference of two square can simplify factoring big numbers, it can also simplify factoring decimals. To demonstrate this, I am going to show the solution to a flamboyantly recondite question.
Now, notice that both numerators simplify via difference of two squares formula.
Thus, the two fractions become
Answer = (D).
Crunching these numbers is cumbersome and, fortunately, unnecessary. Our numerator is a difference of squares, meaning we can break it up by factoring.
This gives us the much simpler
This simplifies to
We can cancel the
Answer = (E).