Answer:
[tex]-\dfrac{463}{7}[/tex]
Explanation:
Given equation: 2x² - 6x - 7 = 0
In quadratic equation: ax² + bx + c
[tex]Sum \ of \ roots : \alpha + \beta = \dfrac{-b}{a}[/tex]
[tex]product \ of \ roots : \alpha \beta = \dfrac{c}{a}[/tex]
So, here given:
[tex]Sum : \alpha + \beta = \dfrac{-(-6)}{2} = 3[/tex]
[tex]Product : \alpha \beta = \dfrac{-7}{2} = - 3.5[/tex]
For finding value:
[tex]\dfrac{\alpha^3}{\beta } + \dfrac{\beta^3 }{\alpha }[/tex]
join fractions
[tex]\dfrac{\alpha^4+ \beta^4}{\alpha \beta }[/tex]
factor out
[tex]\dfrac{(\alpha^2 + \beta ^2)^2 -2\alpha ^2 \beta ^2 }{\alpha \beta }[/tex]
when factored more
[tex]\dfrac{((\alpha + \beta)^2 -2\alpha \beta )^2 -2(\alpha \beta)^2 }{\alpha \beta }[/tex]
insert values inside
[tex]\dfrac{((3)^2 -2(-3.5) )^2 -2(-3.5)^2 }{-3.5 }[/tex]
calculate for value
[tex]-\dfrac{463}{7}[/tex]
