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 product of the roots is given by c/a = 10/1 = 10.

Using Vieta's formulas, the product of the roots is k = 10. Thus, k = 17.

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.

z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - i^2 = 2 + 1 + i = 3 + i.

Let the integers be n and n+1. Then n(n+1) = 240. Solving the quadratic equation n^2 + n - 240 = 0 gives n = 14, so the integers are 14 and 15.

A comprehensive understanding suggests that the lecture was thorough and insightful, making option 2 the correct choice.

The context indicates that the insights were significant and thought-provoking, making 'profound' the correct choice.

The active voice emphasizes the team as the doer, leading to 'The team completed the project.'

The Pulitzer Prize is an American award for achievements in newspaper, magazine and online journalism, literature, and musical composition.

The Pulitzer Prize is awarded for achievements in journalism.

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

For both roots to be negative, the sum of roots (4) must be positive and the product (k) must be positive, hence k > 0.

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

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

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

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.

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

Using the sum of roots: p = -(3 + (-2)) = -1.

The equation factors to (x-1)(x-2) = 0.