If you’re staring at two terms you need to factor, but feel like a deer looking at the headlights of an oncoming semi, here’s a way to leap to safety!
It’s called the “Difference of Two Squares” trick.
It requires four simple steps.
- Figure out if each of the terms is a “perfect square.”
- If so, take the square root of each term.
- Put each square root in its proper place inside two ( ).
- Put a + sign inside the first ( ), and put a – sign inside the second ( ).
Let’s do an easy example. Suppose the terms you’re looking at are these:
x^2 – 9
Let’s go through the 4 steps together.
- Figure out if each term is a “perfect square.”
So, what does it mean for a number or term to be a “perfect square”? It means that you get the number or term by multiplying a number or term by itself. For example, 16 is a perfect square because you can get 16 by multiplying 4 by itself: 4 x 4 = 16.
So when we look at our two terms, x^2 and 9, we notice that both
are perfect squares.
9 is just 3 times 3.
And in the same way, x^2 is just x times x.
- Take the square root of each term.
The square root of x^2 is just x.
And the square root of 9 is just 3.
- Put each square root in the proper place inside two sets of ( ).
We put the square root of the term that was positive first, and the square root of the term that was negative second.Since the x^2 was the positive term, we put its square root, x, first inside each
( ). So far, that gives us: (x ) (x )
Since the 9 was the negative term because it had the negative sign in front of it: – 9, we put its square root, 3, second inside each ( ). So our ( )s now look like this: (x 3) (x 3)
- Finally, we just need to put in signs that connect the terms inside
the ( )s.
That’s easy. We put a + sign inside one ( ), and we put a – sign
inside the other ( ).
I prefer to put the + inside the first ( ), but it really doesn’t matter.The final factored form, then, looks like this: (x + 3) (x – 3)
That’s all there is to it.
Now try these problems for practice.
a) x^2 – 16
b) x^2 – 100
c) x^2 – 121
d) x^4 – 16x^2
e) 49x^8 – 144y^12
a) x^2 – 16 = (x + 4) (x – 4)
b) x^2 – 100 = (x + 10) (x – 10)
c) x^2 – 121 = (x + 11) ( x – 11)
d) x^4 – 16x^2 = (x^2 + 4x) (x^2 – 4x)
e) 49x^8 – 144y^12 = (7x^4 + 12y^6)(7x^4 – 12y^6)