Question

a and b are zeros of f(x)=x^2-p(x+1)-c,show that (a+b)(a×b)=1-c​

Answer 1

Step-by-step explanation:

  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