Question

if a, b,c are in ap then prove that the roots of the equation ax^2 +2bx+ c =0are equal

Answer 1

a , b and c are in GP .
Let a = P/r
b = P
c = Pr
where P is constant and r is common ratio.
now,
ax² +2 bx + c = 0 put here a, b and c value

(P/r)x² + 2Px + Pr = 0
P{ x²/r + 2x + r } = 0
P ≠ 0 so, x²/r + 2x + r = 0

x² +2 xr + r² = 0

now, D = b² -4ac = (2r)² -4.r² = 4r²-4r² =0
D = 0
we know, when D = 0 it means quadratic equation have both roots equal.

hence,
if a , b and c are in GP
then, ax² + 2bx + c = 0 have both roots equal.