Question

If the roots of equation a minus b x square + b minus c x + c minus a is equal to zero are equal then show that c a and b are in ap

Answer 1

Answer:

Step-by-step explanation:

let r & s be the roots, then:

r + s = -(b - c)/(a - b)

but r = s:

2r = -(b - c)/(a - b)

r = -(b - c)/[2(a - b)]

also:

r * s = r^2 = (c - a)/(a - b)

(b - c)^2/[4(a - b)^2] = (c - a)/(a - b)

(b - c)^2/[4(a - b)] = (c - a)

b^2 - 2bc + c^2 = 4ac - 4a^2 - 4bc + 4ab

4a^2 - 4ab + b^2 - 4ac + 2bc + c^2 = 0

(2a - b)^2 - 2c(2a - b) + c^2 = 0

[(2a - b) - c]^2 = 0

2a - b - c = 0

2a = b + c