The Ultimate AP Calculus Quiz!

Explanation
If 4a - 8 > 0 and 4b + 8 < 0, it means that a is greater than 2 and b is less than -2. Therefore, when we multiply a positive number (a) with a negative number (b), the result will always be negative. Hence, ab is negative.

Explanation
The given expression b/b+1 represents the division of b by b+1. Since z is a positive number not equal to 1, b will also be positive. Dividing a positive number by a positive number will always result in a positive value. Therefore, the expression b/b+1 must also be positive.

Explanation
The function f(x) is defined as (10 - x)^2. We are given that y = f(6), which means y = (10 - 6)^2 = 16. We need to find 4y, which is equal to 4 * 16 = 64. To find the corresponding value of x for this result, we need to find the value of x that satisfies f(x) = 64. Evaluating the options, we find that f(18) = (10 - 18)^2 = 64, which matches the desired result. Therefore, the correct answer is f(18).

Explanation
The distance between the b-intercept and the y-intercept of a line can be found by taking the absolute difference between their respective y-coordinates. In this case, the b-intercept is the y-intercept, which is the point where the line crosses the y-axis. To find this point, we set x = 0 in the equation c = 2/3x - 6. Solving for c gives us c = -6. Therefore, the y-intercept is (0, -6). The distance between this point and the b-intercept is the absolute value of -6, which is 6. However, the answer given is √117, which is not equal to 6. Therefore, the given answer is incorrect.

Explanation
The given expression is 3a + 9b + 12. To find an equivalent expression, we can factor out the common factor of 3 from the first two terms, which gives us 3(a + 3b). Then, we add the constant term of 12 to get the final expression of 3(a + 3b) + 12.

Explanation
The function f(x) = √3x - 2 represents a square root function. Since the square root of any number is always non-negative, the smallest possible value of f(x) occurs when √3x - 2 is equal to 0. Solving for x, we get x = 2/3. Therefore, the smallest possible value of f(x) is 0.

Explanation
The equation x^2 - |5x| = -6 can be rewritten as x^2 + 6 = |5x|. Since the absolute value of a number is always non-negative, the right side of the equation must be non-negative as well. This means that x^2 + 6 must be greater than or equal to 0. The only possible value of x that satisfies this condition is x = 2.

Explanation
The given equation states that 3 multiplied by the quantity (a-2) is equal to 24. To find the value of 5 multiplied by the quantity (a-2), we can first solve the given equation. Dividing both sides of the equation by 3 gives us (a-2) = 8. Now, we can substitute this value into the expression 5(a-2). Multiplying 5 by 8 gives us 40, which is the answer.

Explanation
The equation √a + 22 = 38 can be solved by subtracting 22 from both sides to isolate the square root term. This gives us √a = 16. Squaring both sides of the equation eliminates the square root, resulting in a = 256. Therefore, the correct answer is 256.

Explanation
The question states that b can be expressed as 7z + 3, where z is an integer. To find the possible values of b, we need to substitute different values of z and check if the resulting expression falls within the given range of 50 to 70.

For z = 7, 7z + 3 = 52, which falls within the range.
For z = 8, 7z + 3 = 59, which falls within the range.
For z = 9, 7z + 3 = 66, which falls within the range.

Therefore, the possible values of b are 52, 59, and 66.