Learning · Memories

Fun with quadratic equations

The term “quadratic equation” might provoke fear in anyone who didn’t enjoy maths at school. I did enjoy maths at school, more so than my brother or sister, and over the years have used some straightforward multiplications to try and take some of the fear out of quadratic equations.

It started with my older brother, who has lived in Spain since the 1980s. We were sat in a bar 30 minutes from his home with time on our hands. We had made an early start to take his car for a service. I had followed in his wife’s car and it was 20 minutes or more before the garage opened. We were having a chat over coffee and pastries and whatever we were discussing led to a calculation. It was something like 19 times 21 (which is 399) or 24 times 26 (which is 624). Whatever it was I could do it in my head, as I did with the calculations in the previous sentence.

“How did you that?” he asked

“Maths,” I said with the flippancy that comes from knowing him all my life.

“Yes, but how did you work it out?”

We had time for me to explain.

“Let me ask you a question first. What is 10 x 10?”

“100,” he answered.

“Right. 10 x 10 is 100. What’s one more than 10?”


“And what’s one less than 10?”


“Great. Let’s take those two numbers and multiply them. What’s 11 x 9?”


“Yes. And 99 is one less than 100. What’s 5 x 5?”


“Yes. Same thing again, take the numbers either side of it, one more than 5 and one less than 5. What have you got?”

“6 x 4”

“Which is … ?”


“Correct again. 24 is one less than 25, one less than 5 x 5. Let’s do one more for luck. What’s 9 x 9?”


“And one more than 9, and one less than 9, 10 x 8?”


“And 80 is one less than 81. The thing is, there’s a pattern here. If you know what the square of a number is, that is, what happens when it’s multiplied by itself, like 10 x 10, you can work out what happens if you multiply it by the number either side. It’s always going to be the square, minus 1.”

He was engaged. This kind of thing could have prompted a familiar, and maybe happy, feeling from his schooldays, but he didn’t recall learning any of this.

“So,” I continued, “If you know what 20 times 20 is …”


“… yes, you can work out what 21 x 19 is. It’s …”


“Yes. And if you know that 30 x 30 is …”


“Then …”

“29 x 31 is 899”

In later years, when having this sort of conversation with others, I would occasionally say something along these lines to the newly (and maybe only briefly) enthused mathematician: “Wow, you can work out 29 x 31 in your head. Are you Rain Man?”

Back in the bar in Spain we went further.

“How about 10 add 2 multiplied by 10 take away 2?”

“10 add 2, that’s 12, times 10 take away 2, that’s 8 … 96”

“Okay, so that’s 100 take away 4.”

“Why 4? Why not 100 take away 2?”

“Quadratic equations: it’s 10 squared take away 2 squared, not 100 take away 2.”

And I wrote the following on a scrap of paper:

(10 + 2) x (10 – 2)

“Don’t worry, it’s going to be okay. That’s a quadratic equation. If you expand it you get 4 things: 10 x 10, plus 2 x 10, minus 2 x 10, minus 2 x 2.” I drew lines on it to show how it all connected up: 100 + 20 – 20 – 4. “Plus 20 and minus 20 cancel each other out, so it ends up as 10 squared take away 2 squared. Which is 96.”

“Is that what a quadratic equation is?”

“Yes. It’s more scary if it looks like this …”

And I wrote the following:

(a + b)(c – d)

“But all that does is replace numbers with letters, and it still expands like this …”

And I wrote:

ab + bc – bd – ad

“This might be a bit easier …”

And I wrote:

(x + 1)(x – 1) = x2 + x – x – 12 = x2 – 1

“The same principle applies for x + 3 and x – 3 for instance: you get x2 minus 32 which means x2 minus 9. So 17 times 23, which you couldn’t have done five minutes ago, is (20 + 3) times (20 – 3), or 400 minus 9, which is 391. If the numbers you’re multiplying are in this format, the same distance either side of a square, then you might be able to work it out in your head.”

I have no idea if he has used this information in the 20 years or more since we had this conversation. Even if he hasn’t it was a harmless enough way to spend 15 minutes. By now the garage was open and we took his car in for a service, as planned.


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