Question

If `alpha` and `beta` are the zeros of the quadratic polynomial `f(x)=a x^2+b x+c` , then evaluate: (i) `beta/(aalpha+b)+alpha/(abeta+b)` (ii) `a((alpha^2)/beta+(beta^2)/alpha)+b(alpha/beta+beta/alpha)`

Answer 1

`f(x)=ax^2+bx+c`
`alpha+beta=(-b)/a and alphabeta=c/a`
`alpha^2+beta^2=(alpha+beta)^2-2alphabeta=b^2/a^2-2c/a`
1.)`(beta)/(aalpha+b)+(alpha)/(abeta+b)=(a(alpha^2+beta^2)+b(alpha+beta))/(a^2alphabeta+ab(alpha+beta)+b^2)`
substituting values we get=>`-2/a`
2.) `a(((alpha)^3+(beta)^3)/((alphabeta)))+b((alpha^2+beta^2)/(alphabeta))`
`((alpha)^3+(beta)^3)=(alpha^2+beta^2-alphabeta)(alpha+beta)`
substituting values we get=>`b`