Explain FOUR (4) Second Partial Derivatives of the

Two friends wash cars to make extra money. The profit P(x) of one friend after x days can be represented by the function P(x) = −x2 + 9x + 10. The second friend's profit can be determined by the function Q(x) = 12x. Solve the system of equations. What solution is a viable answer to the question, "After how many days will the two students earn the same profit?" and which solution is a nonviable answer? Show your work and justify your answer.

the function y=-16x²+100 represents the hight y(in feet) of a pencil x seconds after falling out the window of a school building. find and interpret the x and y intercepts

The table below shows the probability of a machine manufacturing defective parts in one day.

      Find the variance.

A scuba diver dove from the surface of the ocean to an elevation of −79

feet at a rate of −20 feet per minute. After spending 15 minutes at that elevation, the diver ascended to an elevation of −18

feet. The total time for the dive so far was 22

minutes. What was the rate of change in the diver's elevation during the ascent? Round your answer to the nearest hundredth.

Solving a systemof eqns by given initial conditions
${\displaystyle x^{\prime }\left(t\right)=2x+8y}$
${\displaystyle y^{\prime }\left(t\right)=-x-2y}$
Which has the I.C. ${\displaystyle X\left(0\right)=(6,-2)}$. So I take that it means this:
${\displaystyle 6=2x+8y}$
${\displaystyle -2=-x-2y}$
and then it is solved as any other system? We get eigenvector ${\displaystyle (1,\frac{1}{2})}$, but since the eigenvalues of the system are two, ${\displaystyle \pm 2i}$, we have to find the second generalized eigenvector.
$\left(\begin{array}{cc}2-2i& 8\\ -1& -2-2i\end{array}\right)\left(\begin{array}{c}1\\ \frac{1}{2}\end{array}\right)=\left(\begin{array}{c}6-2i\\ -2-i\end{array}\right)$
So the second vector would be ${\displaystyle (6-2i,-2-i)}$. Using the general solutions for the system, we get
$y\left(t\right)=e^{2it}(1,\frac{1}{2})+e^{-2it}(6-2i,-2-i)$
Would this be correct, or did I misinterpret the initial conditions?

Solving 2nd order homogenous linear ODE with a squared coefficient
I am attempting to solve a differential equation of the form:
${\displaystyle \frac{d^{2}f}{d{x}^2}-{\gamma }^{2}f=0}$
I have set up and solved the characteristic equation as:
$\begin{array}{r}{z}^2-{\gamma }^{2}z+0z=0\\ z(z-{\gamma }^2)=0\end{array}$
which is satisfied when ${\displaystyle z=0}$ and when ${\displaystyle z={\gamma }^2}$
There are two distinct roots (${\displaystyle r_1\text{and}{r}_2}$) hence the general solution should be of the form
${\displaystyle f\left(x\right)=A{e}^{r_{1}x}+B{e}^{r_{2}x}}$
In this case:
$\begin{array}{rl}f\left(x\right)& =A{e}^{0\times x}+B{e}^{{\gamma }^{2}x}\\ & =A+B{e}^{{\gamma }^{2}x}\end{array}$
Apparently, this solution is incorrect as in the answers it is given as:
${\displaystyle f\left(x\right)=A{e}^{\gamma x}+B{e}^{-\gamma x}}$
I would like to know where I went wrong to give me the incorrect solution

One of the steps involved when writing the standard form of a polynomial is completing the ______.