Question

Find the value of k for which x²−k2x−k⁻¹ has real and equal roots.

(a) k=−1

(b) k=0

(c) k=1

(d) k=2

Answer 1

The value of k for which the quadratic equation x² - k²x - k⁻¹ = 0 has real and equal roots is -1, as the discriminant of the equation must be zero for this to occur.

Step-by-step explanation:

The quadratic equation in question is x² - k²x - k⁻¹ = 0.

For a quadratic equation to have real and equal roots, the discriminant (b² - 4ac) must be equal to zero.

In this equation, a = 1, b = -k², and c = -k⁻¹.

Using the formula for the discriminant, we get:

(-k²)² - 4(1)(-k⁻¹) = 0
k⁴ + 4k⁻¹ = 0
k⁵ = -4
k = -4⁻¹⁵
k = -1

Therefore, the value of k for which the quadratic equation has real and equal roots is -1, which corresponds to option (a) k = -1.