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## anonymous one year ago Help MEDAL GIVEN !!!

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1. anonymous

The roots of the equation $2x^2 + 5x - 8 = 0$ are $\alpha$ and $\beta$. Find the quadratic equations whose roots are : $5\alpha + \frac{ 1 }{ \alpha }, 5\beta + \frac{ 1 }{ \beta }$

2. anonymous

@robtobey

3. Jhannybean

Have you found the sum and product of them?

4. anonymous

no

5. anonymous

I did this :

6. anonymous

$\alpha+\beta = \frac{ -5 }{ 2 }$ , $\alpha \beta = \frac{ -8 }{ 2 } = -4$ New sum = $5\alpha + \frac{ 1 }{ \alpha } + 5\beta + \frac{ 1 }{ \beta }$ I don't know what to do now

7. campbell_st

well why not make it $\frac{5\alpha^2 + 1}{\alpha} + \frac{5\beta^2 + 1}{\beta}$

8. Jhannybean

Then you take the sum and product of that?

9. campbell_st

then get the common denominator and add the numerators

10. anonymous

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 :( !!!

11. Jhannybean

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

12. campbell_st

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

13. campbell_st

oops should read $5 \alpha + 5 \beta + \frac{1}{\alpha} + \frac{1}{\beta}$

14. anonymous

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

15. anonymous

so was I correct then ????

16. anonymous

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

17. campbell_st

5(-5/2) + (-5/2)/-4 = -100/8 + 5/8 = -95/8 so that seems correct

18. anonymous

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

19. campbell_st

as for the product I thought $(5\alpha + \frac{1}{\alpha})(5\beta + \frac{1}{\beta} = 25\alpha \beta + \frac{5\alpha}{\beta} + \frac{5\beta}{\alpha} + \frac{1}{\alpha \beta}$

20. campbell_st

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

21. campbell_st

so I think you need to be careful with the signs

22. anonymous

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

23. campbell_st

i got -4

24. anonymous

ok ....

25. campbell_st

so if -b/a = -95/8 then c/a = -32/8 so a = 8, b = 95 and c -32 that's my best guess

26. anonymous

I got something else

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