Quadratic Function Real Life Examples and Applications

Want to know the square root of 7.29 by long division method
a. 3
b. 2.5
c. 2.6
d. 2.7

A boat is traveling east across a river that is 112 meters wide at 8 meters per second. If the river has a northward current of 5 meters per second, what is the resultant speed of the motorboat rounded to the nearest tenth?
8.6 m/s
3.0 m/s
13.0 m/s
9.4 m/s

Find square root of $121$?

Simplify $\frac{4-<!--\; -\; -->\sqrt{7}}{4+\sqrt{7}}$

More precise way of solving inequality
I need to solve this function:
$|<!--\; |\; -->x^2-<!--\; -\; -->1|<!--\; |\; -->\ge <!--\; \ge \; -->2x-<!--\; -\; -->2$
I solved this equation:
For $x<0$, the solution is non existing, here I got negative root, when I tried to solve quadratic function and for $x\ge <!--\; \ge \; -->0$ I got points $x_1=-<!--\; -\; -->1$ and $x_2=3$.
My question is:
How do I set the solution of equation. Is there any procedure, with wich I can determine is equation valid for $[-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->1]\vee <!--\; \vee \; -->[3,+\mathrm{\infty <!--\; \infty \; -->}]$.
I know that I can just set the numbers and see the result, but I just want to now is there any other different way to do this.

Complex Quadratic Programming Problem
I have a quadratic programming problem as follows:
$\underset{x}{min}\Vert <!--\; \Vert \; -->Ax+b{\Vert <!--\; \Vert \; -->}_2$
with no constraints. If A,x, and b are real, It would be a straight forward QP problem that can be also written as follows:
$\underset{x}{min}\frac{1}{2}{x}^{T}Jx+c^{T}x$
where $J=A^{T}A$ and $c=A^{T}b$. Such problems can be handled by Matlab. However, in my case, A, x, and b are all complex, and therefore can't be solved by Matlab (My only tool at the moment). Of course, the target function (the norm) is real.
My question: Any Idea on how to reform such a problem to get a real QP problem so I can use Matlab to solve it?

Determining a component of velocity vector of a fluid flow explicitly
If (u(y),0,0) are the velocity components of an incompressible Newtonian fluid flow due to a pressure gradient in the X-direction, then u(y) is a
(a) linear function of y.
(b) quadratic function of y.
(c) cubic function of y.
(d) constant.
I started with Euler dynamical equation as follows:
$\frac{D\stackrel{\to <!--\; \to \; -->}{q}}{Dt}=\stackrel{\to <!--\; \to \; -->}{F}-<!--\; -\; -->\frac{1}{\mathrm{\rho <!--\; \rho \; -->}}(\mathrm{\nabla <!--\; \nabla \; -->}p)$
where $\stackrel{\to <!--\; \to \; -->}{q}=(u(y),0,0)$ and $\stackrel{\to <!--\; \to \; -->}{F}$ is the external force, $\mathrm{\nabla <!--\; \nabla \; -->}p$ being pressure gradient. Now as per question we have along X-direction that
$$\begin{gathered} \frac{\mathrm{\partial <!--\; \partial \; -->}u(y)}{\mathrm{\partial <!--\; \partial \; -->}t}+u(y)\frac{\mathrm{\partial <!--\; \partial \; -->}u(y)}{\mathrm{\partial <!--\; \partial \; -->}x}+0.\frac{\mathrm{\partial <!--\; \partial \; -->}u(y)}{\mathrm{\partial <!--\; \partial \; -->}y}+0.\frac{\mathrm{\partial <!--\; \partial \; -->}u(y)}{\mathrm{\partial <!--\; \partial \; -->}z}=0-<!--\; -\; -->\frac{1}{\mathrm{\rho <!--\; \rho \; -->}}\frac{\mathrm{\partial <!--\; \partial \; -->}p}{\mathrm{\partial <!--\; \partial \; -->}x} \\ ⟹<!--\; ⟹\; -->\frac{\mathrm{\partial <!--\; \partial \; -->}u(y)}{\mathrm{\partial <!--\; \partial \; -->}t}=-<!--\; -\; -->\frac{1}{\mathrm{\rho <!--\; \rho \; -->}}\frac{\mathrm{\partial <!--\; \partial \; -->}p}{\mathrm{\partial <!--\; \partial \; -->}x} \\ ⟹<!--\; ⟹\; -->u(y)=-<!--\; -\; -->\int <!--\; \int \; -->(\frac{1}{\mathrm{\rho <!--\; \rho \; -->}}\frac{\mathrm{\partial <!--\; \partial \; -->}p}{\mathrm{\partial <!--\; \partial \; -->}x})dt+C \end{gathered}$$
C being arbitrary constant. But from here I am unable to conclude whether this is linear or quadratic or cubic or constant. Is my thinking correct here?

Alternative for calculating the nth of quadratic sequence
Given the quadratic sequence
$f(n)=1,7,19,37,\cdots <!--\; \cdots \; -->$
To calculate the f(n) for $n\ge <!--\; \ge \; -->1$.
$f(n)=a{n}^2+bn+c$
We start with the general quadratic function, then sub in for $n:=1,2$ and 3
$f(1)=a+b+c$
$f(2)=4a+2b+c$
$f(3)=9a+3b+c$
Now solve the simultaneous equations
$\begin{array}{}\text{(1)}& a+b+c=1\end{array}$
$\begin{array}{}\text{(2)}& 4a+2b+c=7\end{array}$
$\begin{array}{}\text{(3)}& 9a+3b+c=19\end{array}$
$(2)-<!--\; -\; -->(1)$ and $(3)-<!--\; -\; -->(2)$
$\begin{array}{}\text{(4)}& 3a+b=6\end{array}$
$\begin{array}{}\text{(5)}& 5a+b=12\end{array}$
$(5)-<!--\; -\; -->(4)$
$a=3$
$b=-<!--\; -\; -->3$
$c=1$
$f(n)=3n^2-<!--\; -\; -->3n+1$
This method is very long. Is there another easy of calculating the f(n)?

Quadratic Bézier curve points
A quadratic Bézier curve is the path traced by the function B(t), given points $P_0$, $P_1$, and $P_2$.
$C(t)=\underset{i=0}{\overset{2}{\sum <!--\; \sum \; -->}}(\genfrac{}{}{0ex}{}{2}{i})t^i(1-<!--\; -\; -->t{)}^{2-<!--\; -\; -->i}{P}_i$
$C(t)=(P_0-<!--\; -\; -->2P_1+P_2)t^2+(-<!--\; -\; -->2P_0+2P_1)t+P_{0}t\in <!--\; \in \; -->[0,1]$
What exactly is $P_0$ or $P_1$ or $P_2$ concerning this equation?
Yes they are points. But in my understanding, a point is a pair of numbers (in 2D-space).
Let $=(1,1)$, $P_1=(1,7)$ and $P_2=(7,1)$.
What values do you use (and where)?
How do you calculate the Bezier Curve for these points?

Determining quadratic function of this word problem
A tour company offers back country hiking excursions at $160 per person forgroups up to 50. For groups larger than 50, the company will reduce the cost of every ticket by $2 for each person in excess of 50. What size of group would produce the greatest revenue?
I'm not worried about finding out what the size of the group is that produces the greatest revenue. What I want to know is what the equation for this is. I was able to figure out this equation:
$f(x)=160x-<!--\; -\; -->2(x-<!--\; -\; -->50),$
where x is the number of people and f(x) is the revenue. But that's a linear function and we're doing quadratic functions. So, There should be an exponent and I'm thinking there's more to this question than I see? Or is it a linear function?

How is discriminant related to real x?
Solve for range of the function,
$y=\frac{x^2+4x-<!--\; -\; -->1}{3x^2+12x+20}$
Text book says, cross multiply and express the obtained equation as a quadratic equation in x
So I get $(3y-<!--\; -\; -->1)x^2+(12y-<!--\; -\; -->4)x+(20y-<!--\; -\; -->1)=0$
Now it says find discriminant D, so we have,
$D=-<!--\; -\; -->4(3y-<!--\; -\; -->1)(8y+3)$
Now it says, set $D\ge 0$ as x is real. Wait what?
Isn't $x\in <!--\; \in \; -->\mathbb{R}$ the domain for a quadratic function? Meaning "x" is always real? What does a discriminant got anything to do with x being real, when all discriminant tells us is whether or not the ROOTS are real? Help please.
To be more specific about my doubt, here's an edit.

I need two numbers that multiply to get -20 and add to get 1. I'm doing a quadratic functions problem and I can't find the answer to this part of the question.
It started out as finding the zero of the equation by factoring. The equation was $g(x)=2x^2+x-<!--\; -\; -->10$. I set it equal to zero, $x^2+x-<!--\; -\; -->20=0$. Now I have to factor but I can't find a way to. If it's prime, I don't know how to answer the main question.
$a\times <!--\; \times \; -->b=-<!--\; -\; -->20$ and $a+b=1$

Solve this quadratic equation by completing the square:
$-2t^2+3t+2=0$

Real world use of even and odd functions
What is the real world use of knowing whether a function is odd or even? Any practical examples?
For example, quadratic equations, differential equations and calculus in General is used for, among other things, determining motion of bodies. Just knowing that a function is symmetrical or not, help how? In terms of solving practical problems with the knowledge.

Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?
The minimal solution (A, B) with respect to some function of A and B, usually $A+B$, is taken. The equation is then rearranged into a quadratic with coefficients in terms of B, one of whose roots is A, and Vieta's formulas are used to determine the other root to the quadratic.
However, I do not fully see the need to take the minimal solution, can someone explain that step for me?

For what class of functions is steepest descent guaranteed to converge in finite number of iterations?
Is there any classification of ${\mathbb{R}}^n\to <!--\; \to \; -->\mathbb{R}$ functions with respect to finite convergence of steepest descent (and similar first-order methods, e.g., nonlinear conjugate gradients)?
By finite convergence I mean that algorithm reaches a point $x_k$ with $||\mathrm{\nabla <!--\; \nabla \; -->}f(x_k)||=0$ after a finite number of iterations, given that no round-off errors occur (and maybe assuming that line search is exact).
It is known that algorithms are globally convergent, i.e. $li{m}_{k\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}||\mathrm{\nabla <!--\; \nabla \; -->}f(x_k)||=0$, but there is no guarantee that it happens for finite k.
For example, conjugate gradient method is guaranteed to converge in n steps for positive definite quadratic functions. Are there any similar results for steepest descent?
What about (strongly) convex general functions? I was only able to find results of the type
$k=O\left(\frac{||x_0-<!--\; -\; -->x^{\ast <!--\; \ast \; -->}|{|}^2}{\mathrm{ϵ<!--\; ϵ\; -->}}\right)$
to achieve $f(x_k)-<!--\; -\; -->f(x^{\ast <!--\; \ast \; -->})\le <!--\; \le \; -->\mathrm{ϵ<!--\; ϵ\; -->}$, where f is smooth convex and x∗ is a stationary point.
As far as I understand, it does not imply finite convergence for $\mathrm{ϵ<!--\; ϵ\; -->}=0$. Is it known that finite convergence does not happen even for strongly convex functions?

Finding the Maximum and Minimum of Trigonometric Equations By Quadratic Substitution
$y=3\cos <!--\; \; -->2x+7\sin <!--\; \; -->x+2$
The original question was: Finding the stationary point of the function. I tried to use both differentiation and trigonometric substitution of cos(2x) into $1-<!--\; -\; -->2{\sin }^2<!--\; \; -->(x)$.
For the quadratic method, I substituted sin(x) as u and solved for u, where the the equation is equated to 0, and finding the midpoint of the roots to find out the value of x following the nature of a parabolic equation.
Unfortunately I realized that trigonometric substitution eliminates a possible solution for x where $\cos <!--\; \; -->(x)=0$. I suspect that it might be due to a non function being converted into a function but I do not quite understand why does it causes so.

Suppose $\gamma$ is a kth root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
Let $k\in <!--\; \in \; -->\mathbb{Z}$ with $k>2$ and suppose $\mathrm{\gamma <!--\; \gamma \; -->}$ is a kth root of unity that satisfies a quadratic equation $z^2-<!--\; -\; -->mz-<!--\; -\; -->n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
My knowledge on algebra and number theory is poor, I just need this lemma to complete the computation of the analytic automorphism groups of complex tori.Can you help me prove this lemma with the knowledge of field theory and Euler phi function as small as possible? Actually I forget almost everything on field theory and have never learned number theory.

Correct notation for inequality equation. I'm currently taking a Polynomial functions -course and learning about quadratic inequality equations.
Let's consider this equation:
$-<!--\; -\; -->x^2+7x<10$
The correct answer to this according to our book is simply
$x<2$ or $x>5$
But is there any other way to express this in a more mathematical way? I understand why this is expressed the way it is as we haven't learned about set theory yet, but I'd like to know how it is expressed "correctly".

I've been using the quadratic equation to factor $2z^2+5iz-<!--\; -\; -->2$. Obviously the roots are $z=-<!--\; -\; -->2i$ and $z=-<!--\; -\; -->i/2$.
Why can't I then write it out as $(z+2i)(z+i/2)$ which is not equivalent to the initial function?