Find k, if the sum of the roots of the quadratic equation 4x2 + 8kx + k + 9 = 0 is equal to their product.

5. Find k, if the sum of the roots of the quadratic equation 4x² + 8kx + k + 9 = 0 is equal to their product.

Sol. 4x² + 8kx + k + 9 = 0

Comparing with ax² + bx + c = 0 we have a = 4, b = 8k, c = k + 9

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

α + β  = α β  ......(i) [Given]

(i.e.) –b/a  = c/a 

∴ ^{- 8k}/⁴ = ^(k+9)/₄

∴ -8k = k+9

∴ -8k – k = 9

∴ - 9k = 9

∴ k = 9/-9

∴ k = -1

Find k, if one of the roots of the quadratic equation kx2 – 7x + 12 = 0 is 3.

3. Find k, if one of the roots of the quadratic equation kx² – 7x + 12 = 0 is 3.

Sol. kx² – 7x + 12 = 0

∵ 3 is one of the root of the quadratic equation, kx² – 7x + 12 = 0

∴  x = 3 satisfies the equation,

Substituting x = 3 in equation we get,

k (3)² – 7 (3) + 12 = 0

∴ 9k – 21 + 12 = 0

∴  9k – 9 = 0

∴  9k = 9

∴  k =  9/9

∴  k = 1

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