Question

if alpha and beta are the zeros of the quadratic polynomial f(x)=x^2-p(x+1)-c,show that (alpha +1) (beta+1)=1-c​

Answer 1

f(x) = x² - p x + q

α and β are the roots of the above equation.

 to find   α² / β² +  β² / α² = ?

  α = [ p + √(p² - 4q)  ]  / 2        and    β = [ p - √(p² - 4 q) ] / 2

so,      α + β = p      and    α β  =  q          and     α² β²  = q²

  =>    α² + β²   =  (α+β)² - 2 αβ  =  p² - 2 q

  =>    α⁴ + β⁴  =  (α² + β²)² - 2 α²β²  =  (p² - 2q)² - 2 q²

                      =  p⁴ - 4 p² q + 4 q² - 2 q²

                      =  p⁴ - 4 p² q + 2 q²

 

NOW ,  α² / β² + β² / α² =  [ α⁴  + β⁴ ] / α² β² =  

                    = [ p⁴ - 4 p² q + 2 q² ] /  q²