Question

Prove that a+c=2a, if the roots of the equation (b-c P+(c-a)x+(a-b),are equal
​

Answer 1

Answer:

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

D=b2−4ac=0

(b−c)x2+(c−a)x+(a−b)=0

Comparing with 

ax2+bx+c=0

Here, a=(b−c), b=(c−a) and c=(a−b)

So,

⇒(c−a)2−4(b−c)(a−b)=0

⇒c2+a2−2ac−4(ab−b2−ac+bc)=0

⇒c2+a2−2ac−4ab+4b2+4ac−4bc=0

⇒c2+a2+2ac+4b2−4ab−4bc=0

⇒(c+a)2+4b2−4b(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