prelim q caringbah '20 (1 Viewer)

[yolo tengo]

hello all, i'm a bit confused with this question, could i have some help with this please? how am i supposed to find the roots? i'm not sure on how to start this question.

 

[howcanibesmarter]

2 methods, either know the exact formula (which u should know if/when u do 4u)

a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-(ab+bc+ac))+3abc

Here since (a+b+c)=0, then we only focus on 3(abc) which is equal to 3(3/2) = 9/2 (This way is faster, however is prone to mistakes since if you memorise it wrong, then you will get 1/3 max, sometimes even 0/3)

Alternatively you can do it this way (this method is what teachers usually want you to do): Since a,b,c are roots, we can rewrite the cubic equation in the following ways.
2a^3+4a-3=0
2b^3+4b-3=0
2c^2+4c-3=0

Adding up these three equations gives

2(a^3+b^3+c^3)+4(a+b+c)-9=0

Rearranging for a^3+b^3+c^3 gives

[9-4(a+b+c)]/2, which is just (9-0)/2 = 9/2, since (a+b+c)=0

I hope that clears it up for u, there are sometimes questions asking for like a^4+b^4+c^4 and to the power of 5, etc, which u cannot use the first method for since the formula is insanely long

See if you can apply the 2nd method into this question

 

[yolo tengo]

2 methods, either know the exact formula (which u should know if/when u do 4u)

a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-(ab+bc+ac))+3abc

Here since (a+b+c)=0, then we only focus on 3(abc) which is equal to 3(3/2) = 9/2 (This way is faster, however is prone to mistakes since if you memorise it wrong, then you will get 1/3 max, sometimes even 0/3)

Alternatively you can do it this way (this method is what teachers usually want you to do): Since a,b,c are roots, we can rewrite the cubic equation in the following ways.
2a^3+4a-3=0
2b^3+4b-3=0
2c^2+4c-3=0

Adding up these three equations gives

2(a^3+b^3+c^3)+4(a+b+c)-9=0

Rearranging for a^3+b^3+c^3 gives

[9-4(a+b+c)]/2, which is just (9-0)/2 = 9/2, since (a+b+c)=0

I hope that clears it up for u, there are sometimes questions asking for like a^4+b^4+c^4 and to the power of 5, etc, which u cannot use the first method for since the formula is insanely long

See if you can apply the 2nd method into this question View attachment 39348

hii thank you sm!! what's the name of the 2nd method if you don't mind me asking?? and yes, i'll give the question a go for sure.

 

[howcanibesmarter]

hii thank you sm!! what's the name of the 2nd method if you don't mind me asking?? and yes, i'll give the question a go for sure.

theres isnt really a name for it, just a method to solve these type of questions

 

[yolo tengo]

theres isnt really a name for it, just a method to solve these type of questions

ohhh haha, thank you

 

[yolo tengo]

See if you can apply the 2nd method into this question View attachment 39348

yo i am practically brain dead atm i feel like i know how to get to the end but its not clicking in my head or maybe i'm doing this wrong

 

[howcanibesmarter]

Heres the solution, if anythings not clear let me know and I can clear it up for u.

 

[nsw..wollongong]

aww u drew a heart

 

[howcanibesmarter]

thats not my working

 

[nsw..wollongong]

thats not my working

sshhh