1.

If `alpha` and `beta` are the roots of `ax^2+bx+c=0`, then the equation `ax^2-bx(x-1)+c(x-1)^2=0` has roots

Answer» As `alpha` and `beta` are the roots of `ax^2+bx+c = 0`.
`:. alpha+beta = -b/a and alphabeta = c/a`
Now, second equation is,
`ax^2-bx(x-1)+c(x-1)^2 = 0`
`=>(a-b+c)x^2-x(b-2c)+c = 0`
Let `m` and `n` are the roots of this equation.
Then, `mn = c/(a-b+c) = (c/a)/(1-b/a+c/a)`
`=(alpha beta)/(1+(alpha+beta)+alpha beta)`
`=(alpha beta)/((1+alpha)+beta(1+alpha))`
`:. mn=(alpha beta)/((1+alpha)(1+beta))->(1)`
Now, `m+n = (2c-b)/(a-b+c) = ((2c)/a - b/a)/(1-b/a+c/a)`
`= (2alpha beta + alpha+ beta)/(1+(alpha+beta)+alpha beta)`
`= (alpha beta + alpha beta+ alpha+ beta)/(1+(alpha+beta)+alpha beta)`
`= (alpha(beta+1)+beta(alpha+1))/((1+alpha)(1+beta))`
`m+n= alpha/(alpha+1)+ beta/(beta+1)->(2)`
From (1) and (2),
`m = alpha/(alpha+1), n = beta/(beta+1)`


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