Ask Uncle Colin: An Absurd Quadratic | Colin

Dear Uncle Colin,

In a recent exam, I was invited to solve $12x^2 – 59x + 72=0$ without a calculator. Is that a reasonable thing to ask?

Very Irate EdExcel-Taught Examinee

Hi, VIETE, and I don’t blame you for being cross – in a non-calculator exam, I’m not sure that really tests the Supposedly Important Skills you’re meant to have.

That said, it’s not impossible.
Method 1: using the formula

This is a bit brutal, but it can be done: if $a=12$, $b=-59$ and $c=72$, you have $\frac{59 \pm \sqrt{(-59)^2 – 4(12)(72)}}{2(12)}$. None of that is especially nice, but $59^2 = 3481$, which…

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An Integral Diversion | Colin

The brilliant @dragon_dodo sent me this puzzle:

Evaluate $\int_0^1 \left(1-x^\frac{1}{7}\right)^3 – \left(1-x^\frac{1}{3}\right)^7 \d x$.

I’m not going to give you the solution right now; that will come after I’ve rambled for a bit.

After I’d solved the puzzle (see below), I wondered what each of the integrals actually evaluated to. Failing to solve it myself immediately, I turned to Wolfram Alpha, which told me it was $\frac{1}{120}$. Interesting. I played with the numbers a bit, and came to the experimental conclusion that $\int_0^1 \left(1 -x^\frac{1}{a}\right)^b \d x =…

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Ask Uncle Colin: Spotting factors | Colin

Dear Uncle Colin,

In a recent test, I stumbled across $9x^4 + \frac{1}{144x^4} + \frac{1}{2}$, which apparently factorises as $\left(3x^2 + \frac{1}{12x^2}\right)^2$. How on earth am I supposed to spot that?!

– Feeling Almost Cheated, That’s Only Reasonable

Hi, FACTOR, and thanks for your message!

I wouldn’t instinctively spot that that factorises — but I would spot that it’s a hot mess.

I’d certainly notice that $9x^4$ is a factor common to the first term and the bottom of the second, and I’d substitute $y = 9x^4$ to see if it made things better: now it’s $y + \frac{1}{16y} +…

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Ask Uncle Colin: An Enormous Sum | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I’ve got a question that asks me to find the coefficient of $x^5$ in $(1+x)^5 + (1+x)^6 + (1+x)^7 + … + (1+x)^{100}$.

I can easily work out the coefficient in each term (it’s just $\nCr{k}{5}$), but I can’t see an easy way to add them up. Any ideas?

– Perhaps A Simple Calculation Answers Logically

Hi, PASCAL, and thanks…

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My Stab At Colin’s Puzzle | Colin

The estimable @colinthemathmo suggests a method for estimating the radius of the earth, which he credits to a sundial expert friend named Mike:

Stand on a wall, perhaps two metres high, and wait for sunrise. When you see the sun just peak above the horizon, start the stopwatch, and jump off the wall (or drop the stopwatch to a friend). Since you are now lower down, you can no longer see the Sun, so wait a bit until you can see it again, and stop the clock.

And astonishingly, that’s enough!

When Colin says “the simplified sums are not too nasty”, you can bet you’re in for an…

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Ask Uncle Colin: An Absolute Quadratic Inequality | Colin

Dear Uncle Colin,

I’m pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help?

– Nasty Absolute Value Inequalities Ending Rongly

Hi, NAVIER, and thanks for your message!

These are nasty, but can be made simpler by rearranging and sketching.

I would begin by getting the absolute value on one side and everything else on the other: $ x^2 – x – 2 < \left| 5x-3\right|$.

The left-hand side is a quadratic…

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Ask Uncle Colin: An Absolute Quadratic Inequality | Colin

Dear Uncle Colin,

I’m pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help?

– Nasty Absolute Value Inequalities Ending Rongly

Hi, NAVIER, and thanks for your message!

These are nasty, but can be made simpler by rearranging and sketching.

I would begin by getting the absolute value on one side and everything else on the other: $ x^2 – x – 2 < \left| 5x-3\right|$.

The left-hand side is a quadratic…

Continue reading at:

http://ift.tt/2tCmlr9