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

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus k = 1.

Using Vieta's formulas, p = -(3 + 4) = -7 and q = 3 * 4 = 12. Therefore, p + q = -7 + 12 = 5.

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

The discriminant is given by b^2 - 4ac. Here, b = 2, a = 1, c = 1, so the discriminant is 2^2 - 4*1*1 = 0.

The sum of the roots is -1 + -3 = -4, and the product is (-1)(-3) = 3. Thus, k = 3.

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

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

The product of the roots ab is given by c/a. Here, c = 6 and a = 1, so ab = 6.

Using the product of the roots, c = 2 * 3 = 6.

For real and distinct roots, the discriminant must be greater than zero: 6^2 - 4*1*k > 0 leads to k < 9.

The discriminant must be greater than zero for real and distinct roots: (-4)^2 - 4*1*k > 0, which simplifies to 16 - 4k > 0, or k < 4.

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

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

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

From Vieta's formulas, p = -sum of roots = -5 and q = product of roots = 6.

The numbers are 4 and 8, as 4 + 8 = 12 and 4 × 8 = 32.

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.

|-3| = 3, so |x| + x = 3 - 3 = 0.

The absolute value |x| of -5 is 5.

x² + y² = 3² + 4² = 9 + 16 = 25.

2x - 3 = 2(4) - 3 = 8 - 3 = 5.

3x - 2 = 3(4) - 2 = 12 - 2 = 10.

|z|² = (1)² + (1)² = 1 + 1 = 2.

z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.

The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

Using Euler's formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.