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The smallest positive value of k for which the equation $x 2 + k x + 9 = 0$ has real roots is

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k=5

k=6

k=3

k=4

answer is B.

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Detailed Solution

Given that the equation

x 2 + k x + 9 = 0

has real roots.
The discriminant

D

of the quadratic equation

a x 2 + b x + c = 0

D = b 2 − 4 a c

, where a, b are coefficients and

c

is constant.
Compare the given equation with the standard form of quadratic equation

a x 2 + b x + c = 0

.
We get

a = 1, b = k, c = 9.

Substitute the above values in the discriminant formula.

D = k 2 − 4 × 1 × 9 D = k 2 − 36 D = (k − 6) (k + 6), For D ≥ 0 as roots are real 0 ≥ (k − 6) (k + 6)

0 ≥ (k − 6) or 0 ≥ (k + 6) 6 ≥ k or − 6 ≥ k we need smallest poitive k . 6 ≥ k

Hence, the smallest positive value of k for which the equation

x 2 + k x + 9 = 0

has real roots is 6.
The correct option is (2).

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