It began with a burning sore throat on Sunday, and soon the familiar trigonometry formulae that she was revising with her students began to blur before her eyes.

Despite some students remarking that she was looking like a ghost ~~and probably hoping under their breath that class would end early~~, Miss Loi managed to finish up the lesson heroically, before canceling classes for the day to rest at home.

Back home, no miracle was forthcoming from the spoonfuls of 京都念慈菴川貝枇杷膏, even though it comes in a lethal blend of 15 herbal ingredients including 川貝、枇杷葉、沙參、茯苓、陳皮、桔梗、法半夏、瓜蔞仁、款冬花、遠志、乾薑、薄荷、蜂蜜 (don’t worry this won’t come out in your exams).

After rolling around in bed for hours, a fever developed. Upon discovering that she had the power to somehow infect her online friends one after another, she decided it was time to visit the neighbourhood doctor to relieve her from her misery.

After rolling around the chair for hours at the clinic (and getting herself exposed to more germs in the process), she finally sat before her doctor who told her in an ominous voice:

My Sexy Maths Tutor, you have been struck by a deadly αβ strain of the flu virus whose

rootshave mutated into α^{2}and β^{2}!

The roots of the viral equation 4`x`^{2} – `x` + 36 = 0 are `α`^{2} and `β`^{2}. For the vaccine to cure you, I’ll need to find

- an equation whose roots are and ,
- two distinct equations whose roots are
`α`and`β`.

Can you help the doctor cure Miss Loi in order to prevent her from infecting and destroying mankind with her virus? *sneeze*

P.S. The events described in this blog post are true, up to the point when the doctor opened his mouth.

With the inclusion of *Sum and Product of Roots* into Quadratic Equations in the New AMaths Syllabus, you can expect a part or two catering to this topic alongside your usual question on quadratic roots and discriminant i.e. your `b`^{2} – 4`a``c`.

While this topic is rather limited in scope i.e. you really only need to know:

For `a``x`^{2} + `b``x` + `c` = 0 with roots `α`, `β`,

- sum: (note the minus sign!)
- product:

and the above equation can be rewritten as:

`x`^{2}– (sum)`x`+ (product) = 0 (note the minus sign AGAIN!)

Do be mindful that the following expressions (which may not be highlighted well enough in your textbook – but you should still be able to derive them easily in times of need) are often needed as well in your calculations:

`α`^{2}+`β`^{2}= (`α`+`β`)^{2}– 2`α``β`- (
`α`–`β`)^{2}= (`α`+`β`)^{2}– 4`α``β`

## 10 Comments

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Umm.... you *did* know I was kidding when I suggested you make a maths problem out of your recent bout with the flu on Plurk, right?

Right?

right?

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永远支持你。miss loi,miss loi number 1

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FoxTwo:... *sneeze*Piak:Thanks for your fervent support *sneeze*曜

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Awww *pat pat* come come I make chicken soup for you....

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* Miss Loi takes a break from her sneezing *

Part 1

The equation is:

It's interesting that you've chosen to use additional notations i.e. letting

α' =α^{2}, Σα' etc. etc.But to keep to our O-Level scope, we're dealing with

quadraticroots (not higher order polynomials) so we're pretty sure that there'll only betwolonely rootsα&β(somehow this reminds me of Wall-E & Eve 🙂 )So for 4

x^{2}-x+ 36 = 0 with rootsα^{2},β^{2}:Sum of roots: ----- (1) (∵ )

Product of roots: ----- (2) (∵ )

For an equation with roots :

Sum of roots:

using (1), (2) above

Product of roots: using (2) above

So the equation for part 1 is

which can also be expressed as 36

x^{2}-x+ 4 = 0 😉P.S.

Someone: These are essentially the same steps you've done but just curious on your choice of notation.Part 2

This is key part of the working. In their haste, many students have forgotten about this expression α

^{2}+ β^{2}= (α + β)^{2}- 2αβ when trying to calculate α, β from the given α^{2}, β^{2}and vice versa!(reject)

Actually you already know at this point that

αβcannot be negative - since is obtained fromαβ= -3 😉because when , is negative

The 2 equations are:

Once again this can also be expressed as: 2

x^{2}± 5x+ 6 = 0 😉Pardon me if I have any errors, I'm doing this late at night, and with NO PAPER, using latex as working. Haha

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By the way, shouldn't:

and the above equation can be rewritten as:

be:

and the above equation can be rewritten as:

since

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Hi Ms Loi, take care and get well soon! 🙂

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Haha. My choice of notation is from the all powerful Bostock and Chandler. Heehee. Yay! I got it correct. Woot.

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I prefer to use

because it helps me to remember the general form of Vieta's forumulas

when you sum , you take the next letter b (taking alpha as the letter a), with a minus sign in front BECAUSE its an ODD number of roots, getting

when you sum , you take the next letter c (taking beta as letter b), without any minus sign in front BECAUSE its an EVEN number of roots getting

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And so it came to pass that on a historic weekend when The Temple reverberated with the sounds of roaring F1 cars, Miss Loi was finally healed by a potent vaccine that was concocted by

Someone, using ingredients from anA-Level武林密籍 that overwhelmingly killed the virus in an instant.P.S. O-Level students are advised not to be alarmed by the

chimconcepts presented in Comment #9, for you'll ONLY be dealing withtworoots in your exam - so just focus on the stuff in the orange boxes within the main blog post.P.P.S. The typo mentioned in Comment #6 has been fixed. Miss Loi was

reallysick when she put out this blog post (yeah yeah excuses excuses ...)Last but not least, thanks

FoxTwoandDanielfor pats pats and your well-wishes. The sneezing has completely stopped now 😀