Question

If the roots of equation (a-b)x²+(b-c)x+(c-a)=0 are equal then prove that 2a=b+c.

Answer 1

When the roots of the equation (a-b)x² + (b-c)x + (c-a) are equal
→ (b−c)² −4 (a−b) (c−a) =0 
→ b²+c²−2bc −4 (ac−a²−bc+ab) =0 
→ b²+c²−2bc −4 ac+4a²+4bc−4ab =0 
→ b²+c²+2bc −4 ac+4a²−4ab =0 
→ b²+c²++4a² +2bc −4 ac−4ab =0 
→ (b+c −2 a)² =0 
→ (b+c −2 a)=0

→ b+c=2a