Question

If the roots of the polynomial (b-c)x^2+(c-a)x+(a-b)=0 has equal roots then prove that 2b=a+c

Answer 1

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

D = b² - 4ac

equation (b-c)x²+(c-a)x+(a-b)=0

here,

a = (b-c)

b = (c-a)

c = (a-b)

so,

D = (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(a+c)(2b) = 0

[(c+a)-(2b)]² = 0

(c+a)-2b = 0

c+a = 2b