Would you like to make your high-school math homework a little easier? Read on.
This post is a little technical, but stick with it. If you are a high-school student, you will be glad you did.
The Difference-of-Squares Formula
Every algebra student encounters a chapter on "factoring quadratics." Somewhere in that chapter is a formula for factoring a "difference of squares."A difference of squares is just what it sounds like: something to the second power subtracted from something else to the second power. Another way to say "to the second power" is "squared." So X² - Y² is one example of a difference of squares.
The formula for factoring a difference of squares is
X² - Y² = (X + Y) × (X – Y)
As I said, every algebra student learns this formula.
Most of them promptly forget the formula as soon as the test is over. That is a mistake.
The Pythagorean Theorem
Let us now briefly turn out attention to the famous Pythagorean Theorem. In words, the Pythagorean Theorem says that “the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs.” The hypotenuse is the longest side of the right triangle. The legs are the two shorter sides. Here is the Pythagorean Theorem as a formula.H² = A² + B²
From the time a student first encounters the Pythagorean Theorem, they see it again and again until they have taken their last math class. Unless they choose a profession that uses math. In that case, they see the Pythagorean Theorem again and again until they retire.
The Best Kept Secret
What does the Pythagorean Theorem have to do with the “difference of squares” formula? That, my friends, is the best kept secret in math.Of course, it is not really a secret. It is just something wonderful that is rarely mentioned in text books and rarely taught in class. Don’t ask me why.
From Algebra I and Geometry onward, and on SAT and ACT tests, students often encounter problems that require them to find one leg of a right triangle given the hypotenuse and the other leg. For instance, the student is told that a particular right triangle’s hypotenuse is length 5 and one of its legs is length 3. The problem is to find the length of the other leg. The student rightly recognizes that this is a job for the Pythagorean Theorem.
The theorem tells the student that 5² = 3² + B². The student rearranges this equation to say that B² = 5² - 3². And most students simply do the calculations just that way. B² = 25 – 9 = 16.
But take another look at the expression 5² - 3². It is a difference of squares. So 5² - 3² is the same as (5 + 3) × (5 – 3). That is, 5² - 3² = (5 + 3) × (5 – 3) = 8 × 2 = 16.
Either way, the length of the other leg is the square root of 16, which is 4.
With these simple numbers, the difference-of-squares approach does not look that much easier than the ordinary approach. But what if the hypotenuse is 25 and the given leg is 24? Which calculation is easier?
B² = 25² - 24² = 625 – 576 = 49
B² = (25 + 24) × (25 – 24) = 49 × 1 = 49
The best kept secret in math is that the difference-of-squares formula is actually useful. In particular, the difference-of-squares formula makes it easier to find one leg of a right triangle if the student is given the hypotenuse and the other leg.
Go now, and do your homework more quickly and easily. And amaze your friends! Teach them what you learned in this post.
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Kip is the owner and Center Director of Mathnasium of Fountain Valley.
Contact him at (714) 593-1500 or email him at FountainValley@Mathnasium.com