Ask Uncle Colin: A factorising trick | Colin

Dear Uncle Colin,

How would you factorise $63x^2 + 32x – 63$? I tried the method where you multiply $a$ and $c$ (it gives you -3969) – but I’m not sure how to find factors of that that sum to 32!

Factors Are Troublesomely Oversized, Urgh

Hi, FATOU, and thanks for your message!

When the numbers in a quadratic like this get large (in mental arithmetic terms, at least), I try not to think about the number itself, but about its prime factorisation.

In this case, we know -3969 is $-63 \times 63$, making it $-3^2 \times 7 \times 3^2 \times 7$ or $-3^4 \times 7^2$.
How does that help?…

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Wrong, But Useful: Episode 62 | Colin

In this month’s episode of Wrong, But Useful, we’re joined by @ch_nira, who is Dr Nira Chamberlain in real life – and the World’s Most Interesting Mathematician.
Nira is a professional mathematical modeller, president-designate of the IMA, and a visiting fellow at Loughborough university.
We discuss Nira’s entry in the Aperiodical Maths-Off. I mention a simulation of the Schelling model.
Number of the podcast: 2, the percentage of the top rate of National Insurance.
The Big MathsJam Special and Big MathsJam itself.
Dave has a circle question that stumped his students. Nira turned the…

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Bone Marrow Odds | Colin

I don’t remember doing it – although I’d meant to for some time – but apparently I signed up for the British Bone Marrow Registry. (If you’re between 17 and 40, you can sign up the next time you give blood; the more people on the register, the more likely it is for people who need a transfer to find a match.)

But this post isn’t about how lovely and generous I am1. Instead, it’s about a stat in the letter they sent me: in any given month, there is a 1-in-240 chance of a given person on the register being matched with someone who needs a donation.
Is that a big number?

For me, that’s…

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Ask Uncle Colin: A Mess of Logs | Colin

Dear Uncle Colin,

I have to show that \(-\frac{x}{2} = \ln (\sqrt{1+e^x} – \sqrt{e^x }) + \ln (\sqrt{1+e^{-x}} + 1)\). I can’t get it anywhere near the right form!

– Proof Of It Not Coming – Any Reasonable Explanation?

Hi, POINCARE, and thanks for your message!

That’s a bit of a mess – but with some careful book-keeping, it comes out ok!

Let’s start by combining the logarithms to get \(\ln\br{\br{ \sqrt{1+e^x} – \sqrt{e^x }}\br{\sqrt{1+e^{-x}}+ 1}}\).

Now we have something inside the log-bracket we can expand to get: \(\sqrt{ \br{1+e^{x}} \br{1+e^{-x}} } + \sqrt{1+e^x} – \sqrt{…

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Exchange rates on board | Colin

“Three teas, please,” said the passenger ahead of me in the queue. The Armorique was due in Plymouth any minute, and tea was of the essence.

“That’s £4.65, or €5.601.”

Hang on a moment, I thought, remembering to order my own tea as well. 560 isn’t a multiple of 3. What’s going on?

“That’s £3.10, please, or €3.75.” That’s not even, either!

I didn’t ask – tea, as I mentioned, was of the essence.

It was obvious (to me) that the prices were being calculated in sterling, then converted into euros – but a moment’s calculation suggested that there was also some rounding going on,…

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Ask Uncle Colin: A Cubic That Won’t Come Good | Colin

Dear Uncle Colin,

I’m told that \(x\sqrt{x} – 5\sqrt{x} = 2\) and I have to find \(x – 2\sqrt{x}\). Everything I try seems to make it worse! Any ideas?

Mastering A Cubic – Help Is Needed

Hi, MACHIN, and thanks for your message! At first glance, that’s a strange one. We can solve it, but I’m not quite sure a) what technique the question would like you to use, or b) whether my approach is the best one. So, with that in mind, let’s have at it!
What’s ugly?

I don’t like the look of those square roots, so let’s let \(y = \sqrt{x}\) and turn it all into a cubic: \(y^3 – 5y = 2\). That…

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