Problem Let \alpha and \beta be the solutions of the quadratic equation 2x^2-6x-7=0. Find the value of (\alpha^3)/(\beta)+(\beta^3)/(\alpha).

Question

Problem

Let
`\alpha` and
`\beta` be the solutions of the quadratic equation
`2x^2-6x-7=0`. Find the value of
`(\alpha^3)/(\beta)+(\beta^3)/(\alpha)`.

Answer 1

`-(463)/(7)`

Step-by-step explanation:

Given equation: 2x² - 6x - 7 = 0

In quadratic equation: ax² + bx + c

`Sum \ of \ roots : \alpha + \beta = (-b)/(a)`

`product \ of \ roots : \alpha \beta = (c)/(a)`

So, here given:

`Sum : \alpha + \beta = (-(-6))/(2) = 3`

`Product : \alpha \beta = (-7)/(2) = - 3.5`

For finding value:

`(\alpha^3)/(\beta ) + (\beta^3 )/(\alpha )`

join fractions

`(\alpha^4+ \beta^4)/(\alpha \beta )`

factor out

`((\alpha^2 + \beta ^2)^2 -2\alpha ^2 \beta ^2 )/(\alpha \beta )`

when factored more

`(((\alpha + \beta)^2 -2\alpha \beta )^2 -2(\alpha \beta)^2 )/(\alpha \beta )`

insert values inside

`(((3)^2 -2(-3.5) )^2 -2(-3.5)^2 )/(-3.5 )`

calculate for value

`-(463)/(7)`