If One Root of the Quadratic Equation ax 2 bx c 0 is Equal to n

Answer: Sol. Let α, β be the roots of eq. ax2 + bx + c = 0

According to the question,

Β = αn

Also α + β = -b/a ; αβ = c/a

α β = c/a ⇒ α. α n = c/a ⇒ α = (c/a)1/n + 1

then α + β = – b/a ⇒ α + αn = -b/a

or (c/a)1/n + 1 + (c/a)n/n + 1 = -b/a

⇒ a .(c/a)1/n + 1 + a .(c/a)n/n + 1 + b = 0

⇒ an/n + 1 c1/n + 1 + a1/n + 1 c n/n + 1 + b = 0

⇒ (an c) 1/n + 1 + (can) 1/n + 1 + b = 0

Hence proved.