Question

Help MEDAL GIVEN !!!

Answer 1

The roots of the equation \[2x^2 + 5x - 8 = 0\] are \[\alpha \] and \[\beta \]. Find the quadratic equations whose roots are :

\[5\alpha + \frac{ 1 }{ \alpha }, 5\beta + \frac{ 1 }{ \beta } \]

Answer 2

@robtobey

Answer 3

Have you found the sum and product of them?

Answer 4

no

Answer 5

I did this :

Answer 6

\[\alpha+\beta = \frac{ -5 }{ 2 } \] , \[\alpha \beta = \frac{ -8 }{ 2 } = -4\]


New sum = \[5\alpha + \frac{ 1 }{ \alpha } + 5\beta + \frac{ 1 }{ \beta } \]



I don't know what to do now

Answer 7

well why not make it

\[\frac{5\alpha^2 + 1}{\alpha} + \frac{5\beta^2 + 1}{\beta}\]

Answer 8

Then you take the sum and product of that?

Answer 9

then get the common denominator and add the numerators

Answer 10

then for new product: \[(5\alpha + \frac{ 1 }{ \alpha })* (5\beta+\frac{ 1 }{ \beta })\]


in the end I got the answer to be : \[8x^2 -95 - 786\]. which I believe is wrong :( !!!

Answer 11

then input the sum and product into \(\sf x^2 -(\text{sum of roots})x +(\text{product of roots})=0\)?

Answer 12

look at the sum of the roots this way


\[5\alpha + 5 \beta = \frac{1}{\alpha} + \frac{1}{\beta} = 5(\alpha + \beta) + \frac{\alpha + \beta}{\alpha \beta}\]

does that help

Answer 13

oops should read

\[5 \alpha + 5 \beta + \frac{1}{\alpha} + \frac{1}{\beta}\]

Answer 14

ahh yes that does help very much .. cheers :)

Answer 15

so was I correct then ????

Answer 16

what did you guys get as your final answer???

Answer 17

5(-5/2) + (-5/2)/-4 = -100/8 + 5/8 = -95/8

so that seems correct

Answer 18

what about the product ... it seems way tooo big

Answer 19

as for the product

I thought

\[(5\alpha + \frac{1}{\alpha})(5\beta + \frac{1}{\beta} = 25\alpha \beta + \frac{5\alpha}{\beta} + \frac{5\beta}{\alpha} + \frac{1}{\alpha \beta}\]

Answer 20

which becomes

\[25 \alpha \beta + \frac{5 \alpha^2 + 5\beta^2}{\alpha \beta} + \frac{1}{\alpha \beta}\]

or

\[25 \alpha \beta + \frac{5[(\alpha + \beta)^2 - 2 \alpha \beta]}{\alpha \beta} + \frac{1}{\alpha \beta}\]

Answer 21

so I think you need to be careful with the signs

Answer 22

tooo confusing wht do you get as finl answer ???

Answer 23

i got -4

Answer 24

ok ....

Answer 25

so if -b/a = -95/8 then c/a = -32/8

so a = 8, b = 95 and c -32

that's my best guess

Answer 26

I got something else