The product of the roots is given by c/a = 6/1 = 6.

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

For both roots to be positive, k must be greater than 9 (since the sum of roots is -b/a and must be positive).

For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.

The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.

Using the sum of roots formula, p = -(3 + (-2)) = -1. Therefore, p = 1.

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

Factoring gives (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

The condition for equal roots is given by the discriminant: b^2 - 4ac = 0.

Using Vieta's formulas, k = (-1)(-2) = 2.

Using the formula for the sum of roots, -2 + -2 = -4, and product of roots, (-2)(-2) = 4, we find k = 4.

Using the sum of roots, b = - (4 + (-1)) = -3.

For the roots to be equal, the discriminant must be zero, which means b^2 = 4ac.

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

Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.

The product of the roots is given by c/a, which is 9/3 = 3.

The product of the roots is given by c/a = 6/1 = 6.

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

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

Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0.

The equation can be factored as (2x - 3)(2x - 3) = 0, indicating a perfect square.

The discriminant is 3^2 - 4*5*(-2) = 9 + 40 = 49, which is a perfect square, hence the roots are rational.

The equation can be factored as (x + 3)^2 = 0, hence a = 3.

This is a perfect square trinomial: (x + 3)(x + 3) = 0.

For equal roots, b^2 - 4ac = 0. Here, 6^2 - 4(1)(k) = 0, so k = 9.

For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.

The equation can be factored as (x - 2)(x - 2) = 0, which is (x - 2)^2.

The equation can be factored as (x - 3)(x - 3) = 0, or (x - 3)^2 = 0.

The equation can be factored as (x - 3)^2 = 0, hence a = 3.