1.3.4 Sum of Squares: Factoring Method

Usually on the 3rd of 4th column of the test you will have to compute something like: $(30^2 - 2^2) + (30 + 2)^2$ (with the subtracting and additions might be reversed). Instead of memorizing a whole bunch of formulas for each individual case, it is probably just best to view these as factoring problems and using the techniques of FOILing to aid you. So for our example:

$$(30^2 - 2^2) + (30 + 2)^2 = 2 \cdot 30^2 + 2 \cdot 30 \cdot 2 + 2^2 - 2^2 = 1800 + 120 = 1920$$

Usually the number needing to be squared is relatively simple (either ending in 0 or ending in 5), making the computations a lot easier. Other times, another required step of converting a number to something more manageable will be necessary. For example:

$$19^2 + (10^2 - 9^2) = (10 + 9)^2 + (10^2 - 9^2) = 2 \cdot 10^2 + 2 \cdot 10 \cdot 9 + 9^2 - 9^2 = 200 + 180 = 380$$

or, if you have your squares memorized and noticed you also have a difference of squares (Section 1.3.6):

$$19^2 + (10^2 - 9^2) = 361 + (10 - 9) \cdot (10 + 9) = 361 + 19 = 380$$

The following are some more problems to give you practice with this technique:

Problem Set 1.3.4