MaxWong

\(4^x - 2^x = 56 + 11\cdot 2^x + 2^{x - 1}\)

If we let \(t = 2^x\) we have 

\(t^2 - t = 56 + 11t + \dfrac12 t\\ 2t^2 - 25t - 112 = 0\\ (t - 16)(2t + 7) = 0\)

It is not possible that 2^x = -7/2, therefore 2^x = 16, which corresponds to x = 4.

Pick a pair of equations and solve the system of those equations, and you will get the vertices.

\(\begin{cases}y = -x-5\\x=2\end{cases} \implies (x, y) = (2, -7)\)

\(\begin{cases}y = -x-5\\y = -\dfrac15x + \dfrac{13}5\end{cases}\implies (x, y) = \left(-\dfrac{19}2, \dfrac{9}2\right)\)

\(\begin{cases}y = -\dfrac15 x + \dfrac{13}5\\x=2\end{cases} \implies (x, y) = \left(2, \dfrac{11}5\right)\)

We find the center of the circle by solving the system 

\(\begin{cases}(x - 2)^2 + (y + 7)^2 = \left(x + \dfrac{19}2\right)^2 + \left(y - \dfrac92\right)^2\\ \left(x + \dfrac{19}2\right)^2 + \left(y - \dfrac92\right)^2 = (x - 2)^2 + \left(y - \dfrac{11}5\right)^2\end{cases}\)

which just says the squared distance from the center to each vertex is the same.

Solving gives the center coordinates \(\left(-\dfrac{49}{10}, -\dfrac{12}5\right)\).

Radius of circle is just the distance between \(\left(-\dfrac{49}{10}, -\dfrac{12}5\right)\) and any one of the vertices, i.e., \(\sqrt{\left(-\dfrac{49}{10}-2\right)^2+\left(-\dfrac{12}5-\dfrac{11}5\right)^2} = \dfrac{23\sqrt{13}}{10}\).

We use center-radius form of circle equation, and we have the answer: \(\left(x + \dfrac{49}{10}\right)^2 + \left(y + \dfrac{12}5\right)^2 = \dfrac{6877}{100}\)

If you want a simpler answer, multiply both sides by 100 and get \((10x + 49)^2 + (10y + 24)^2 = 6877\).

Exercise: Fill in the missed details in my solution.

The exterior angles of the polygon form an arithmetic sequence. 

Then the interior angles of the polygon should also form an arithmetic sequence.

The first term of that arithmetic sequence is 136 and the last term is 145, and there are k terms.

Suppose that the common difference is d degrees. Then \(136 + (k - 1) d = 145\).

That means \((k - 1)d = 9\). Then k - 1 is a factor of 9. Therefore, k can only be 2, 4, or 10.

Obviously, 2-sided polygons are not a thing, and the interior angles are too large for a quadrilateral.

Then k = 10.

You can try to figure out how to present this in a rigorous way.

Where is a?

Anyways, when x approaches -3, the value of f(x) should approach -3 - 6 = -9 from the right.

If f(x) is continuous, then the value of f(x) should approach -9 from the left too.

Then 

\(\displaystyle\lim_{x\to -3} (3x + b) = -9\\ -9 + b = -9\\ b = 0\)

\(\begin{cases}a^3 + 3ab^2 = 679\\3a^2 b + b^3 = -679\end{cases}\)

Adding gives 

\(a^3 + 3a^2 b + 3ab^2 + b^3 = 0\\ (a + b)^3 = 0\)

So a + b = 0.

Just expand it:

(2u^2 - v^3)^4 = (2u^2)^4 - 4(2u^2)^3 (v^3) + 6(2u^2)^2 (v^3)^2 - 4(2u^2) (v^3)^3 + (v^3)^4

The bold term is the term in u^2 v^9. You can just simplify it and you will see that the coefficient is -8.

Generally, to draw a vector (x, y), pick a starting point (a, b), draw an arrow from (a, b) to (a + x, b + y).

c is the vector representing a + b.

What is P?

We need to know a and n first before we can find c.

Note that \(\begin{cases}an = -20\\\dfrac{a^2 n(n - 1)}2 = 180\end{cases}\). (Use binomial theorem and compare coefficients to get this system of equations.)

Solving gives a = -2 and n = 10.

Now, note that \(c = \dfrac{a^3 n(n-1)(n-2)}{6}\). Substitute the values of a and n and simplify.