Raymond Ho's answer to Vignesh anand's Secondary 3 A Maths Singapore question.
Thank u very much. I deeply appreciate it. Thanks again
Vignesh, for the guessing you can just use your in-built calculator function. Also, you must be prepared that the questions can require topics from more than one chapter.
For some reason I can’t see the working fully (it got cut off halfway when I view it from my phone).
For some reason I can’t see the working fully (it got cut off halfway when I view it from my phone).
a³ = 4 - 3a
a³ + 3a - 4 = 0
let f(a) = a³ + 3a - 4
Sub a = 1,
f(1) = 1³ + 3(1) - 4
= 1 + 3 - 4
= 0
By the factor theorem, (a - 1) is a factor of f(a)
So (a - 1)(a² + a + 4) =0
You can easily guess the coefficients of a for the quadratic factor since we know only a² x a = a³ and only 4 x -1 = -4.
Otherwise, use long division
a = 1 or a² + a + 4 = 0
For a² + a + 4 = 0,
Discriminant (b² - 4ac)
= 1² - 4(1)(4)
= 1 - 16
= -15 < 0
Since discriminant < 0, there are no real roots for this part
∴ a = 1
You have to show there are no real roots
a³ + 3a - 4 = 0
let f(a) = a³ + 3a - 4
Sub a = 1,
f(1) = 1³ + 3(1) - 4
= 1 + 3 - 4
= 0
By the factor theorem, (a - 1) is a factor of f(a)
So (a - 1)(a² + a + 4) =0
You can easily guess the coefficients of a for the quadratic factor since we know only a² x a = a³ and only 4 x -1 = -4.
Otherwise, use long division
a = 1 or a² + a + 4 = 0
For a² + a + 4 = 0,
Discriminant (b² - 4ac)
= 1² - 4(1)(4)
= 1 - 16
= -15 < 0
Since discriminant < 0, there are no real roots for this part
∴ a = 1
You have to show there are no real roots
Thank u very much
@ J, I find your method much easier to understand for finding the value of a
@ Eric, u r right . I used the calculator and put a^3+3A-4. I got a=1 and other 2 roots r not real. Thanks
@ Raymond, thanks a lot for all the help
Thanks guys
My calculator model is Casio FX 96 SG Plus. There is an in built function there MODE —> 3 —> 4: aX3 + bX2 + cX + d or something like that. Any non-real roots will have a letter “lowercase i” at the end. So if you ever see two “i” (they come in twos for cubic functions), these are not real.
Yes. I used to have that model. Now using Casio fx-97SG X. This one can find roots until x to the power of 4.
Do note that while you may use the calculator to find the roots, the working still has to be shown as above.
Sure. Will do that. Thanks a lot for all the guidance
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