The discriminant must be less than zero: k^2 - 4*1*16 < 0 leads to k < -8.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.

For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.

For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.

The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

The discriminant is 0, indicating that the roots are real and equal.

If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.

The roots can be found by factorization: (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0. Solving gives k = -8.

The sum of the roots is 3 + (-1) = 2, hence p = -2.

Substituting x = 2 into the equation gives 2^2 - 4*2 + k = 0, which simplifies to k = 4.

Using the root 3 in the equation: 3^2 - 7*3 + k = 0, we get k = 6.

By Vieta's formulas, the sum of the roots p + q = -b/a.

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.

The discriminant is 0, indicating that the roots are real and equal.

The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.

The product of the roots is 1 * (-3) = -3, hence k = -3.

For both roots to be negative, k must be greater than 0.

For both roots to be positive, k must be less than 4 (from the condition of the sum and product of roots).

The discriminant is 0, indicating one real root (a repeated root).