Probability of not occurring = 1 - P(event) = 1 - 0.3 = 0.7.

The probability of either A or B occurring is P(A) + P(B) - P(A and B) = 0.2 + 0.5 - (0.2 * 0.5) = 0.7.

P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

For independent events, P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

For mutually exclusive events, P(C or D) = P(C) + P(D) = 0.2 + 0.3 = 0.5.

The probability that it will not rain is 1 - 0.7 = 0.3.

6 and 7 are consecutive integers whose product is 42, not 72.

The product of 6 and 7 is 42, while 8 and 9 gives 72. Therefore, the correct pair is 8 and 9.

The other number can be found by dividing the product by the known number: 144 / 12 = 12.

If the product is 48 and one number is 6, then the other number must be 48 ÷ 6 = 8.

All pairs listed multiply to 48, hence they are valid pairs.

The other number is 72 divided by 8, which equals 9.

For the product to be a multiple of 15, at least one of the numbers must be a multiple of 5.

Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.

By Vieta's formulas, the sum of the roots p + q = -b/a.

According to Vieta's formulas, the sum of the roots p and q is -b/a and the product is c/a.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.

Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.

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

The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.

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

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

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

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

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

The product of the roots is 1 * (-3) = -3, hence k = -3.

For both roots to be negative, k must be greater than 0.

For both roots to be negative, k must be greater than 0.