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A Quick Way to Solve 1000^2 − 999^2: The Difference of Two Squares

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications and fun mathematical facts.

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Speeding Up Your Arithmetic

Suppose you needed to do the sum 10002 − 9992. What would you do? Pull out a calculator? Panic even? No need. While writing this sentence I quickly calculated in my head that the answer is 1999.

But how did I manage that so quickly? I'm not a human calculator and I don't know all of the squares off the top of my head. Instead I used a very simple, quick trick to solve this seemingly difficult sum. This trick is called the difference of two squares.

What is the Difference of Two Squares Trick?

First of all, let's look at this algebraically. Our sum is of the form a2 - b2, where a and b are both integers (whole numbers). An interesting property of a2 - b2 is that it can be factorised into two brackets like this:

a2 − b2 = a2 − ab + ab − b2

= a(a − b) + b(a − b)

= (a + b)(a − b)

Using this, we can quickly speed up our sum.

10002 − 9992 = (1000 + 999)(1000 − 999)

And there we have it. A seemingly difficult calculation is turned into something very simple and easy to do mentally.

= 1999 × 1

= 1999

How To Find the Difference of Two Squares on the DoingMaths YouTube Channel

Generalising Our Result

We can use this trick to quickly solve any subtraction using consecutive squares.

If we take a perfect square, n2, and the next perfect square, (n+1)2 for some integer n, then:

(n+1)2 − n2 = ((n + 1) + n) ((n + 1) − n)

= (2n + 1) × 1

= 2n + 1

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So, using this result, we can calculate that:

302 − 292 = 2 × 29 + 1

= 59

or

9502 − 9492 = 2 × 949 + 1

= 1899

and so on for any consecutive perfect squares that we like.

Another Use of the Difference of Two Squares For Quick Arithmetic

We can also use the difference of two squares to quickly complete the multiplication of two numbers (as long as their average can be easily squared).

For example, take 38 × 42. The average of these two numbers is 40, which can be easily squared to 1600. We can then use the difference of two squares as such:

38 × 42 = (40 − 2)(40 + 2)

= 402 − 22

= 1600 − 4

= 1596

We can also use it to quickly calculate 105 × 95:

105 × 95 = (100 + 5)(100 − 5)

= 1002 − 52

= 10000 − 25

= 9975

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 David

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