If α, β are roots of ax2+bx+c=0 then the equation ax2−bx(x−1)+c(x−1)2=0 has roots

Since α, β are roots of ax2+bx+c=0∴ α+β=−b/a, αβ=c/a.The equation ax2−bx(x−1)+c(x−1)2=0 can be written as x2(a−b+c)+x(b−2c)+c=0Let, γ,δ be its roots. Thenγ+δ=−(b−2c)a−b+c=−b+2ca−b+c=−ba+2ca1−ba+ca⇒γ+δ=α+β+2αβ1+α+β+αβ=αα+1+ββ+1and, γδ=ca−b+c=ca1−ba+ca=αβ1+α+β+αβ=αα+1⋅ββ+1Thus, the equation ax2−bx(x−1)+c(x−1)2=0 hasγ=αα+1 and δ=ββ+1 as its two roots.