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

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.

The discriminant is 0, indicating one distinct real root.

The discriminant is 0, indicating that there is exactly one distinct real root.

For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.

Let the roots be r and r+2. Then, r + (r+2) = 6 and r(r+2) = k. Solving gives k = 10.

The range of sin^(-1)(x) is [-π/2, π/2].

Range = Maximum - Minimum = 9 - 1 = 8.

Range = Maximum value - Minimum value = 30 - 10 = 20.

The absolute value function has a minimum value of 0, hence the range is [0, ∞).

The range of y = sin^(-1)(x) is [-π/2, π/2].

The rate law for a reaction is determined experimentally and is dependent on the mechanism of the reaction.

The rate law for a reaction depends on the mechanism of the reaction.

The overall order of the reaction is the sum of the powers of the concentration terms, which is 1 + 2 = 3.

This is an example of Van 't Hoff rule, which states that the rate of reaction doubles for every 10°C increase in temperature.

The rate of reaction is expressed in terms of concentration change per unit time, hence the unit is mol/L·s.

This relationship is known as the rate law of the reaction.

Fourier's Law of Heat Conduction states that the rate of heat transfer by conduction is directly proportional to the area and the temperature gradient.

The rate of heat transfer by conduction is directly proportional to the temperature difference between the two ends.

The real part of z = 4 - 3i is 4.

Restoring force F = -kx, where k is the spring constant and x is the displacement.

The right-hand rule can be used to determine the direction of the magnetic field, current, and force in electromagnetic situations.

The kinetic energy per molecule is given by KE = (1/2) * m * v^2. Thus, KE = 0.5 * m * (250)^2.

Speed of sound = sqrt(γRT/M) = sqrt(1.4 * (300^2)) = 400 m/s.

The speed of sound in a gas is given by c = sqrt(γRT/M). For an ideal gas, c is approximately 1.4 times the RMS speed.

Using the quadratic formula, x = [4 ± √(16 - 8)] / 4 = [4 ± 2√2] / 4 = 1 ± √2/2. Hence, the roots are not simple fractions.

Using Vieta's formulas, sum of roots = 1 + 2 = 3 = -(-4)/2 => k = 2*1*2 = 4.

Using the quadratic formula, the roots are x = [20 ± √(400 - 300)] / 10 = [20 ± 10] / 10 = 3 and 1.