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

Using Vieta's formulas, n = 3 * 4 = 12.

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

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.

The equation can be expressed as (x - 3)^2 = 0, which expands to x^2 - 6x + 9 = 0. Thus, k = 9.

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

By Vieta's formulas, ab = 10, which is the constant term of the polynomial.

By Vieta's formulas, the sum of the roots a + b + c = -(-3) = 3.

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.

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

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

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

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

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

The sum of the roots is -p = 8, hence p = -8.

The sum of the roots is given by -(-3)/1 = 3. Thus, k = 3 - 3 = 0.

The sum of the roots is given by -(-6)/1 = 6. The product of the roots is k, and since the sum is correct, k can be any value.

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.

1/x = 1/(1/2) = 2.

x² + y² = 3² + 4² = 9 + 16 = 25.

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

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

Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.

The modulus of z = -1 + i is |z| = √((-1)² + 1²) = √(1 + 1) = √2. The square of the modulus is |z|² = 2.

The real part of a complex number z = a + bi is a. Here, the real part is -3.

|z|² = (1)² + (1)² = 1 + 1 = 2.