Question

If a^2-5a+1=0
then find the value of 1) a^3+a^-3 2)a^2+a^-2

Answer 1

Find the value of 'a' from the first equation by factorising or quadratic formula than sub... it in the next two equations

Answer 2

well that is one way. I'm sure there is a better way to solve the problem, probably using some algebraic formulas.

Answer 3

I go nowhere, hihihi,,, but just want to give out my idea
\(a^2 = 5a -1\\a^3 = 5a^2 -a\) and combine them as you wish.

Answer 4

where do you go after that?

Answer 5

Is it not that \(a^3+a^{-3} = a^3 + \dfrac{1}{a^3}\)??

Answer 6

exactly it is.

Answer 7

so, just put my a^3 in, hihihi

Answer 8

i understood that. The problem arises when I try to simplify things beyond that. Any clues on that?

Answer 9

maybe we can try using symmetry to ease arithmetic :

just notice that if \(\large \alpha \) is a root of given quadratice, then \(\large \dfrac{1}{\alpha }\) is also a root.

Answer 10

ok.. go on

Answer 11

or use a^3 + b^3

Answer 12

next observe that \(\large a^2 + a^{-2} = \left(a + a^{-1} \right)^2 - 2 \)

Answer 13

(a+b)^3-3ab(a+b)

Answer 14

yes that is quite right.

Answer 15

you know sum and product of roots , by vieta's formula

Answer 16

that is , -b/a and c/a

Answer 17

well , i have not tried , maybe 90% it would work

Answer 18

brilliant ! you have just read out my mind :) it works nicely

Answer 19

Eureka ! thanks

Answer 20

say the roots of given quadratic are \(\large a, a^{-1}\) :

\(\large a + a^{-1} = 5\)
\(\large aa^{-1 } = 1\)

\(\large \implies a^2 + a^{-2} = \left(a + a^{-1} \right)^2 - 2 = 5-1 = 4 \)

you can work out \(\large a^3+a^{-3}\) as suggested by @No.name

Answer 21

Well if you know vieta's relation(this relation is different) , you can very well compute a^10 also
i learnt the method a couple of does back it is really a nice one

Answer 22

*\(\large \implies a^2 + a^{-2} = \left(a + a^{-1} \right)^2 - 2 = 5-2 = 3\)

Answer 23

It is similar to theory of equations
well @deeprony7 you got it

Answer 24

would you mind sharing the vieta's relation that you mentioned? @No.name

Answer 25

i watched a video , i will share , give me a moment

Answer 26

the ans to a^2+a^-2 is actually 23(atleast thats what it says in my book) @ganeshie8

Answer 27

https://www.youtube.com/watch?v=y8sQBGK9yYM

Answer 28

Well that is a cubic

Answer 29

Ahh yes, was just checking whether you're really paying attention or not :P
there is a mistake in calculation, haven't u noticed i forgot to square

Answer 30

corrected below\[\large \implies a^2 + a^{-2} = \left(a + a^{-1} \right)^2 - 2 = 5^2-2 =2 3\]

Answer 31

You may not learn that method now

Answer 32

I think that video is talking about Newton's symmetric sums
http://www.artofproblemsolving.com/Wiki/index.php/Newton's_Sums

Answer 33

Oh , someone told me its some vieta thing , lol

Answer 34

It's Newton's sum i c