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 roots have mutated into α2 and β2!
The roots of the viral equation 4x2 – x + 36 = 0 are α2 and β2. 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 b2 – 4ac.
While this topic is rather limited in scope i.e. you really only need to know:
For ax2 + bx + c = 0 with roots α, β,
- sum: (note the minus sign!)
and the above equation can be rewritten as:
- x2 – (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:
- α2 + β2 = (α + β)2 – 2αβ
- (α – β)2 = (α + β)2 – 4αβ