Proof 101

So I’m kinda bored at work today – it’s slow, I’m caught up, and all my projects are in stand-by mode, awaiting action from others. I’m playing around on the scientific calculator on the PC when this pops into my head –

When you square a number and subtract one from the result, is it always the same as if you multiply one less than the original number with one more?

For example,

(30 x 30) – 1 = 899
and
29 x 31 = 899

Or

(4733 x 4733) – 1 = 22,401,288
and
4732 x 4734 = 22,401,288

Is this true in all instances?

Intuitively I could see it was so, but how would you prove it?

Okay, math whizzes step aside. I’m coated with nearly two decades of rust in the math department. It’s hard to flex the math muscles; I’ve probably regressed to somewhere near early high school, depending on the type of high school you’re talking about.

But to quote a co-worker of mine, “I’m proud of myself!” I did it! I proved that the above holds true in all cases, and it only took me a couple of minutes.

How?

It’s easy!

The first step is to translate everything into variables. So the two equations become

x^2 – 1 = y
and
(x – 1) (x + 1) = y

Set them equal to each other and you get

x^2 – 1 = (x – 1) (x + 1)

It’s now a simple algebraic equation. Expand, solve, and you find

x^2 – 1 = x^2 – x + x – 1
x^2 – 1 = x^2 – 1

So, yes, it’s true in ALL cases.

Like the Tin Man, I feel like suddenly I’m walking on the Yellow Brick Road, all squeaks and dings, but at least I’m moving!

Now, for homework –

Can the cube of a number less one ever be equal to three numbers around the original number, multiplied together?

Hmmm?