For no real roots, the discriminant must be less than zero: k^2 - 4*1*9 < 0, thus k > 6.

For one real root, the discriminant must be zero: k^2 - 4*1*9 = 0 => k^2 = 36 => k = ±6.

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

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

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0 => k^2 = 64 => k = 8 or k = -8.

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.

Using Vieta's formulas, k = 3 + 4 = 7.

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.

For d orbitals, the azimuthal quantum number l is equal to 2.

log2(8) = 3 because 2^3 = 8.

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

log_10(0.1) = log_10(10^-1) = -1.

log_10(1000) = 3 and log_10(0.01) = -2, thus 3 - 2 = 1.

log_10(1000) = log_10(10^3) = 3.

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

log_2(32) = 5 and log_2(4) = 2. Therefore, 5 - 2 = 3.

log_2(32) = 5 and log_2(8) = 3. Therefore, 5 - 3 = 2.

Since 32 = 2^5, log_2(32) = 5.

log_2(8) = 3 and log_2(4) = 2, thus log_2(8) + log_2(4) = 3 + 2 = 5.

log_3(27) = 3 and log_3(9) = 2. Therefore, 3 - 2 = 1.

Since 27 = 3^3, log_3(27) = 3.

log_3(81) = log_3(3^4) = 4.

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

log_4(64) = log_4(4^3) = 3.