Find k, if the roots of the quadratic equation x2 + kx + 40 = 0 are in the ratio 2 : 5.

2. Find k, if the roots of the quadratic equation x² + kx + 40 = 0 are in the ratio 2 : 5.

Sol. x² + kx + 40 = 0

Comparing with ax² + bx + c = 0 we have a = 1, b = k, c = 40

Let α  and β  be the roots of given quadratic equation.

Ratio of  α to β  is 2 : 5 [Given]

Let the common multiple be m

∴ α  = 2m and β  = 5m

∴ α +β = ^{- b}/_a = -k/1 = - k

∴ 2m + 5m = - k

∴ 7m = -k

∴ k = -7m

Also, α .β = c/a = 40/1 = 40

∴ 2m× 5m = 40

∴ 10m² = 40

∴ m² = 40/10

∴ m² = 4

∴ m = ±√4

∴ m = ±2

But, k = -7m

∴ k = -7(2)  or -7(-2)

∴ k = -14 or 14