Ok so there is two "puzzles" that I am being asked to solve, here are the directions: Puzzle 1: Use the digits -9 to 9 (only once) to create a real solution between 30 and 80. Show your work to justify your answer. And then the second attached image labeled image 1 - complex puzzles. Puzzle 2: Use the digits -9 to 9 to complete the puzzle below. Try all the combinations. But as you begin doing that, you will realize that in some cases you are looking for factor combinations with a particular sum or difference. You will see that some numbers have to be greater than a particular value in order to produce the product you are looking for. Show your work to confirm your solution. And then the first attached image labeled image 2 - complex puzzles. And finally a reflection: With regard to the two puzzles, please comment about the process of multiplying complex numbers. With regard to: i^n did you observe it while solving the two puzzles, and if yes, how did you compute its value? Thanks in advance for any answers.

Step-by-step explanation:

Puzzle 1

(a + bi) (c + di)

ac + adi + bci + bdi²

(ac − bd) + (ad + bc)i

Matching coefficients:

30 < ac − bd < 80

ad + bc = 0

We need to pick four integers between -9 and 9 such that these two equations are satisfied.  One possible combination is:

a = 8, b = -4, c = 6, d = 3

The number would be:

ac − bd = (8)(6) − (-4)(3) = 60

Puzzle 2

Using the result from Puzzle 1:

ac − bd = 34

ad + bc = 8

Like before, it makes sense to assume b is negative.  With some trial and error, one possible answer is:

a = 5, b = -6, c = 2, d = 4

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.