Question

Problem

Answer 1

Answer:

$-\dfrac{463}{7}$

Explanation:

Given equation: 2x² - 6x - 7 = 0

In quadratic equation: ax² + bx + c

$Sum \ of \ roots : \alpha + \beta = \dfrac{-b}{a}$

$product \ of \ roots : \alpha \beta = \dfrac{c}{a}$

So, here given:

$Sum : \alpha + \beta = \dfrac{-(-6)}{2} = 3$

$Product : \alpha \beta = \dfrac{-7}{2} = - 3.5$

For finding value:

$\dfrac{\alpha^3}{\beta } + \dfrac{\beta^3 }{\alpha }$

join fractions

$\dfrac{\alpha^4+ \beta^4}{\alpha \beta }$

factor out

$\dfrac{(\alpha^2 + \beta ^2)^2 -2\alpha ^2 \beta ^2 }{\alpha \beta }$

when factored more

$\dfrac{((\alpha + \beta)^2 -2\alpha \beta )^2 -2(\alpha \beta)^2 }{\alpha \beta }$

insert values inside

$\dfrac{((3)^2 -2(-3.5) )^2 -2(-3.5)^2 }{-3.5 }$

calculate for value

$-\dfrac{463}{7}$