Question

if the root of the equation( B - C) X^2 +( C- A )X + A -B are equal then proove that 2B=A+C

Answer 1

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

D=b2−4ac=0D=b2−4ac=0

(b−c)x2+(c−a)x+(a−b)=0 ≡ ax2+bx+c=0(b−c)x2+(c−a)x+(a−b)=0 ≡ ax2+bx+c=0

Here, a =(b-c) , b = (c-a) and c = (a-b)

So, D = (c−a)2−4(b−c)(a−b)=0(c−a)2−4(b−c)(a−b)=0

c2+a2−2ac−4(ab−b2−ac+bc)=0c2+a2−2ac−4(ab−b2−ac+bc)=0

c2+a2−2ac−4ab+4b2+4ac−4bc=0c2+a2−2ac−4ab+4b2+4ac−4bc=0

c2+a2+2ac+4b2−4ab−4bc=0c2+a2+2ac+4b2−4ab−4bc=0

(c+a)2+4b2−4b(a+c)=0(c+a)2+4b2−4b(a+c)=0

(c+a)2+(2b)2−2⋅(c+a)⋅(2b)=0(c+a)2+(2b)2−2⋅(c+a)⋅(2b)=0

[(c+a)−(2b)]2=0[(c+a)−(2b)]2=0

⇒ c+a−2b=0⇒ c+a−2b=0

a + c = 2b

Hence proved!