Question

If a, b, c € R show that the roots of the following equation (a-b)x^2 + (b+c-a)x -c=0 are rational.​

Answer 1

Answer:

3x

2

−2x(a+b+c)+(ab+bc+ca)=0

Let D be the discriminant

D=4(a+b+c)

2

−12(ab+bc+ca)

=4[(a+b+c)

2

−3(ab+bc+ca)]

=2[2a

2

+2b

2

+2c

2

−2ab−2bc−2ca]

=2[(a−b)

2

+(b−c)

2

+(c−a)

2

]

If D=0, then

(a−b)

2

+(b−c)

2

+(c−a)

2

=0

⇒a−b=0;b=c;c=a

or a=b=c