The sum of the roots is given by -b/a = 12/3 = 4.

For no real roots, the discriminant must be less than 0: k^2 - 4*1*16 < 0, which gives k < 8.

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

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

The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.

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

The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).

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

The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.

The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.

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

Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.

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

The product of the roots is n = 2 * 3 = 6.

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

The equation can be factored as (x-5)^2 = 0, hence the double root is x = 5.

Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.

The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.

Let the roots be r and r+2. Then, r + (r+2) = 4 and r(r+2) = k leads to k = 4.

For both roots to be positive, k must be greater than 6, as the sum of the roots must be positive.

The discriminant must be negative: k^2 - 16 < 0, hence k > 4.

Substituting x = 2 into the equation gives 2^2 - 3*2 + p = 0 => 4 - 6 + p = 0 => p = 2.

Using the root, we substitute: 2^2 - 6*2 + k = 0 => 4 - 12 + k = 0 => k = 8.

Using Vieta's formulas, if one root is 3, the other root is 7 - 3 = 4. Thus, k = 3 * 4 = 12.

Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.

Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.

Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

Using Vieta's formulas, k = (-2)(-4) = 8.