The discriminant must be non-negative: (2k)^2 - 4(1)(16) ≥ 0. This gives 4k^2 - 64 ≥ 0, leading to k^2 ≥ 16, so k ≥ 4.

The discriminant must be less than zero. Thus, k^2 - 4*1*16 < 0 leads to k^2 < 64, giving k < 8 and k > -8.

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.

For real and distinct roots, the discriminant must be positive: k^2 - 4(1)(16) > 0. Thus, k^2 > 64, leading to k < -8 or k > 8.

For real and distinct roots, the discriminant must be positive: k^2 - 4(1)(16) > 0. Thus, k^2 > 64, leading to k < -8 or k > 8.

For both roots to be negative, k must be negative and |k| > 8. Thus, k = -8.

Using Vieta's formulas, the sum of the roots is 0, so k = -0 = 0. The product of the roots is 9, confirming k = 6.

For both roots to be negative, k must be positive and k^2 > 4(1)(9). Thus, k > 6.

For both roots to be positive, k must be negative and k^2 > 36. Thus, k < -6.

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 equal, the discriminant must be zero. Thus, (-5)^2 - 4(1)(k) = 0. Solving gives k = 6.25.

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.

log_10(0.01) = log_10(10^-2) = -2.

log_2(1/8) = log_2(2^-3) = -3.

log_4(16) = log_4(4^2) = 2.

log_5(1) = 0 because 5^0 = 1.

log_5(25) = 2, so log_5(25) + log_5(4) = 2 + log_5(4).

The 3rd term in the expansion of (a + b)^6 is given by C(6, 2) * a^4 * b^2 = 15 * a^4 * b^2.

The 3rd term in the expansion of (x + 2)^5 is C(5, 2)(x)^2(2)^3 = 10 * x^2 * 8 = 80.

The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.

The coefficient of x^0 in (x + 5)^3 is given by 3C0 * (5)^3 = 1 * 125 = 125.