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The positive value of k for which the equation $x 2$ + kx + 64 = 0 and $x 2$ – 8x + k = 0 will both have real roots, is

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answer is D.

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Detailed Solution

Given: The equations

x 2 + k x + 64 = 0

and

x 2 − 8 x + k = 0

The quadratic equation of the form

a x^{2} + bx + c = 0

have equal roots if the discriminant (D) is equal to zero.
We know,

D = b 2 − 4 a c = 0 … .. (1)

x 2 + k x + 64 = 0; a = 1, b = k, c = 64

Substitute the values in equation (1)

k 2 − 4 × 1 × 64 = 0

⇒ k 2 − 256 = 0 ⇒ k 2 = 256 ⇒ k = ± 16...... (2)

x 2 − 8 x + k = 0; a = 1, b = − 8, c = k

Substitute the values in equation (1)

⇒ (− 8) 2 − 4 × 1 × k = 0 ⇒ 64 − 4 k = 0 ⇒ 4 k = 64 ⇒ k = 644 ⇒ k = 16..... (3)

Thus, from equation (2) and (3):

k = 16

Therefore, the positive value of k for which the equation

x 2 + k x + 64 = 0

and

x 2 − 8 x + k = 0

will both have real roots, is 16.
The correct answer is (4).

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