Voldemort

Draw a line from point O to point C.

Because OB and OC are radii, OB = BC = OC = 4. Therefore, we know that triangle OBC is an equilateral triangle. Solving for the area of OBC gives us 4sqrt(3). 

In addition, since we know that Angle COA is 120 degrees(as angle BOC is 60), the rest of the area would be 1/3 of the area of the whole circle. This is 16/3pi. 

Hence, the area of the shaded region is: 4sqrt(3) + 16/3pi

ab = 64sqrt(3)/3

There is a way to find the exact value! :D

x^3 + 3x^2 + 3x + 1 = 2

(x+1)^3 = 2

x = \(\sqrt[3]{2}-1\)

log_b(5^2) = 2log_b(5) = 28

The domain will be all real numbers except for 2 values. Can you find them? (Hint; They will be a and b!)

The range will also be all real numbers as well except for one value. Try inverting the x and y values and isolating the new y value! (Hint: that number would be c!)

The dot product of u and v is equal to ||u||* ||v||*cos 60 =  3. (Prove this before you start using it!)

row_1(Au) = \(\mathbf{u \cdot u}\) = 9

row_2(Au) = Dot product of u and v, equaling 3

Au = <9, 3>

Similarly:

row_1(Av) = 3

row_2(Av) = v \cdot v = 4

Av = <3, 4>

P(z) = z^2 + bz + c = (z - (-7 + 2i))(z - (-1 - 2i))

P(1) = b + c + 1 = (7 - 2i)(2 + 2i)

14 + 14i - 4i + 4 - 1 

17 + 10i.

I believe that each pair of complex roots are conjugates of each other, no?

But think about it, if they are conjugates, they are mirrors across the x axis. If the distance between two roots conjuagates of each other are different from the other pair of roots which are conjugates, then this would form a trapezoid. Could you see what I mean? Can you continue from here?

What if some fo the roots are real, however? If the roots are real, what does z have to be? Is there even a root that is real? I have to go real soon, so Ill try to post the solution when I am back. 

Use polynomial construction!
S_1 = 3

S_2 = 3

S_3 = 3

We can then construct a polyoimial where a, b, c are roots of this polynomial.

P(x) = x^3 - 3x^2 + 3x - 3

= (x - 1)^3 - 2

(x-1)^3 = 2

Since I believe a has to be real based on the fact that b = d + ei and c = d - ei , \(a = \sqrt[3]{2} + 1\)