The discriminant is 0, indicating that there is exactly one distinct root.

For both roots to be positive, k must be greater than 1 (from Vieta's formulas).

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

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

The equation can be factored as (x - 3)(x - 3) = 0, indicating it has one distinct root (a double root).

Factoring the equation gives (x - 2)(x - 5) = 0, so the roots are 2 and 5.

The polynomial can be expressed as (x - 1)^3, indicating that the root x = 1 has a multiplicity of 3.

The given polynomial can be factored as (x - 1)^3 = 0, which has one distinct real root, x = 1.

The product of the roots is given by c/a. Here, c = 1 and a = 2, so the product is 1/2.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 2 and a = 2, so the product is 2/2 = 1.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 6 and a = 2, so the product is 6/2 = 3.

The sum of the roots is -1 + (-3) = -4, so -2 = -4, which is correct. The product of the roots is (-1)(-3) = 3, so k = 3.

The sum of the roots is -1 + (-2) = -3, and the product is (-1)(-2) = 2. Thus, k = 2.

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.

The discriminant must be greater than zero: (-2)^2 - 4*1*k > 0, which simplifies to k < 1.

For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4*1*k = 0 leads to 25 - 4k = 0, giving k = 6.25.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 6 and a = 2, so the product is 6/2 = 3.

The sum of the squares of the roots is given by (sum of roots)^2 - 2(product of roots). Here, sum = 5, product = 6. So, 5^2 - 2*6 = 25 - 12 = 13.

For the roots to be equal, the discriminant must be zero. Thus, k^2 - 4*1*16 = 0, which gives k^2 = 64, so k = ±8.

For equal roots, the discriminant must be zero. Thus, k^2 - 4*1*4 = 0, which gives k^2 = 16, so k = ±4.

For both roots to be negative, k must be negative and greater than the square root of the discriminant. Thus, k < -6.

The sum of the roots is 3 + 3 = 6, so k = 6.

The sum of the roots is 3 + 3 = 6, so k = 6.

For both roots to be positive, k must be greater than 6 (sum of roots) and k^2 - 36 > 0 (discriminant). Thus, k > 6 and k < 12.

The sum of the roots is -2 + -2 = -4, so k = 4.

The discriminant must be less than zero: k^2 - 4*1*9 < 0 leads to k^2 < 36, hence k < 0 or k > 0.

For the roots to be real and distinct, the discriminant must be positive: k^2 - 64 > 0, leading to k > 8 or k < -8.

The sum of the roots is 2 + 4 = 6, so k = 6.

The sum of the roots is 3 + 3 = 6, so k = 6.

For the roots to be real and distinct, the discriminant must be greater than zero: k^2 - 4(1)(9) > 0, which simplifies to k^2 > 36, hence k < -6 or k > 6.