Question

If the quadratic equation ( b - c) x2 + (c-a) x + ( a - b) = 0 has equal
roots, then show that 2 b = a + c.​

Answer 1

Step-by-step explanation:

If roots of a quadratic equation are equal, then discriminant of the quadratic equation is 0

D=b

2

−4ac=0

(b−c)x

2

+(c−a)x+(a−b)=0

Comparing with

ax

2

+bx+c=0

Here, a=(b−c), b=(c−a) and c=(a−b)

So,

⇒(c−a)

2

−4(b−c)(a−b)=0

⇒c

2

+a

2

−2ac−4(ab−b

2

−ac+bc)=0

⇒c

2

+a

2

−2ac−4ab+4b

2

+4ac−4bc=0

⇒c

2

+a

2

+2ac+4b

2

−4ab−4bc=0

⇒(c+a)

2

+4b

2

−4b(a+c)=0

⇒(c+a)

2

+(2b)

2

−2(c+a)(2b)=0

⇒[(c+a)−(2b)]

2

=0

⇒c+a−2b=0

⇒2b=c+a