Verified answer

Step-by-step explanation:

Since α and β are the roots of ax

2

+bx+c=0,

therefore

α+β=−

a

b

,αβ=

a

c

The equation ax

2

−bx(x−1)+c(x−1)

2

=0 can be written as x

2

(a−b+c)+(b−2c)x+c=0

Sum of the roots of this equation is

S=−

a−b+c

b−2c

=

a−b+x

−b+2c

=

1−

a

b

+

a

c

−

a

b

+

a

2c

⇒S=

1+α+β+αβ

α+β+2αβ

=

α+1

α

+

β+1

β

Product of the roots =

a−b+c

c

⇒P=

1−

a

b

+

a

c

a

c

⇒P=

1+α+β+αβ

αβ

=

α+1

α

.

β+1

β

Thus, ax

2

−bx(x−1)+c(x−1)

2

=0

has

α+1

α

,

β+1

β

as its two roots.

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