Quadratic Function Real Life Examples and Applications

How can we integrate a function for surface area of revolution?
$S=2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_0^8(-<!--\; -\; -->\frac{1}{24}{x}^2+\frac{2}{3}x)\sqrt{1+(-<!--\; -\; -->\frac{1}{12}x+\frac{2}{3}{)}^2}dx$
This is the equation (with my values plugged in) for surface area of revolution which I found online and I understand the derivation, thus I wanted to use it for a math project I am currently working on. I attempted to use the quadratic function (outside the radical) and inside the radical is the first derivative squared. However, I am stuck in evaluating the integral. I tried u substitution where u is equal to $-<!--\; -\; -->1/12x+2/3$ but was unable to finish it.
Is trigonometry required and does the u substitution I attempted not work in this case? What is the Surface area (answer to the integral)?

Rewriting a quadratic Matrix equation as a quadratic vector equation
Consider the set of $N\times <!--\; \times \; -->N$ matrices $\{W_i{\}}_{i=1}^{i=L}$, set of $N\times <!--\; \times \; -->1$ vectors $\{g_i{\}}_{i=1}^{i=L}$ and $\{h_i{\}}_{i=1}^{i=L}$. Now consider the following sum
$\begin{array}{r}S=\underset{i}{\sum <!--\; \sum \; -->}\underset{j}{\sum <!--\; \sum \; -->}{g}_i^H{W}_i{h}_i{h}_j^H{W}_j^H{g}_j\end{array}$
where the summation runs through all L for all i,j. Clearly, this equation is quadratic in the matrix variables $\{W_i{\}}_{i=1}^{i=L}$. Now define the column vector
$\begin{array}{r}w=\left[\begin{array}{c}\mathrm{vec}<!--\; \; -->(W_1)\\ \mathrm{vec}<!--\; \; -->(W_2)\\ ⋮<!--\; ⋮\; -->\\ \mathrm{vec}<!--\; \; -->(W_L)\end{array}\right]\end{array}$
where for a matrix A, vec(A) denotes the column vector containing the columns of A starting from the first column. The question is, can we write
$\begin{array}{r}S=w^{H}Qw\end{array}$
where Q is a matrix which is a function of $\{g_i{\}}_{i=1}^{i=L}$ and $\{h_i{\}}_{i=1}^{i=L}$. If so, what is the structure of Q?

How to evaluate a function with a quadratic argument or higher? How to proceed?
example if I have a function of the form
$$f(2x+5)=4x-<!--\; -\; -->3$$
and I am asked to evaluate
f(8)
I make
$$t=2x+5$$
from where
$$x=(t-<!--\; -\; -->5)/2$$
then
$$f(t)=4((t-<!--\; -\; -->5)/2)-<!--\; -\; -->3$$
I take
$$t=8$$
and that's it.**
Doubts
but how do I do it here, if I do the above I get a quadratic equation or a cubic one
a) $f(x^2+1/x^2)=x+1/x$
find
$$f(4)=?$$
$x>0$ b)
$$f(x^3+1/x^3)=x+1/x$$
find
$$f(4)=?$$
$x>0$

Does a pointwisely convergent sequence of quadratic functions pointwisely converges to a quadratic one?
Let $\{f_j{\}}_{j=1}^{\mathrm{\infty <!--\; \infty \; -->}}$ be a sequence of functions $f_j:{\mathbb{R}}^n\to <!--\; \to \; -->\mathbb{R}$ quadratically parameterized as $f_j(x)=x^T{P}_{j}x$ for a symmetric matrix $P_j\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$. Let $f_j$ pointwisely converges to a function f.
Then, can we say that f is also quadratic so that $f(x)=x^{T}Px$ for some symmetric $P\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$?
I was particularly interested in the case: fj is monotonically increasing and bounded by a quadratic function $g(x)=x^{T}Qx$:
$$f_1(x)\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->f_j(x)\le <!--\; \le \; -->f_{j+1}(x)\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->g(x)<\mathrm{\infty <!--\; \infty \; -->}\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->{\mathbb{R}}^n.$$
In that case, by MCT, $f_j\to <!--\; \to \; -->f$ pointwisely for some f, and we also have $P_j\to <!--\; \to \; -->P$ for some symmetric $P\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, by MCT again, due to
$$P_1\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->P_j\le <!--\; \le \; -->P_{j+1}\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->Q.$$
In this case, $f(x)=x^{T}Px$ is true since $x^T{P}_{j}x\to <!--\; \to \; -->x^{T}Px$ for all $x\in <!--\; \in \; -->{\mathbb{R}}^n$. Moreover, if $f(x)=x^{T}Px$ is true, then it is trivially continuous, so that the convergence is uniform on any compact $\mathrm{\Omega <!--\; \Omega \; -->}\subset <!--\; \subset \; -->{\mathbb{R}}^n$ by Dini's theorem.
However, I can't see whether $f(x)=x^{T}Px$ is true or not in general case without monotonicity.

Turning the two solutions of a quadratic equation in a general solution
So while I was working on a project for uni, I came across this quadratic equation:
$$t^2+2\mathrm{\tau <!--\; \tau \; -->}t-<!--\; -\; -->1=0$$
The solutions for this particular equation are:
$$t_{1,2}=\mathrm{\tau <!--\; \tau \; -->}\pm <!--\; \pm \; -->\sqrt{{\mathrm{\tau <!--\; \tau \; -->}}^2+1}$$
I also found somewhere on the internet that the solutions can be written as one single solution namely:
$$t=\frac{sign(\mathrm{\tau <!--\; \tau \; -->})}{|\mathrm{\tau <!--\; \tau \; -->}|+\sqrt{{\mathrm{\tau <!--\; \tau \; -->}}^2+1}}$$
But I didn't quite understand how they got to that conclusion, is this something that can be done with every quadratic equation with two solutions or is it only for this particular equation?
Perhaps I should also mention that the equation always has a solution between -1 and 1 and that the solution with the sign function always gives the solution that is between -1 and 1.
(The aforementioned equation is an equation that is used to compute givens-rotations, don't know if it is of any importance for the solution of the equation)

Directrix and focus of $a{x}^2+bx+c$
How can you find the directrix and focus of a parabola (quadratic function)
$$a{x}^2+bx+c,$$
where $a\ne <!--\; \ne \; -->0?$ I mean, given the focus x,y and directrix (I'll use a horizontal line for simplicity) $y=k$ you can find the equation of the quadratic; how do you do this backwards?

Existence of a non-square integer L such that L is a quadratic residue modulo $p^n$ for all n.
The background of this question is on the sequences in p-adic integers, in which I originally looked into the following problem:
Can a sequence of perfect squares converge to a non-square value under the p-adic metric?
Where the p-adic metric is, as usual, defined as
$$d_p(m,n)=p^{-<!--\; -\; -->max\{n\in <!--\; \in \; -->\mathbb{N}:<!--\; :\; -->p^n|(m-<!--\; -\; -->n))\}}$$
For $m\ne <!--\; \ne \; -->n$, and $d_p(m,m):=0$
Obviously, it is ask for the existence of a function $f:\mathbb{N}\to <!--\; \to \; -->\mathbb{Z}$ and a non-square number L such that for all $n\in <!--\; \in \; -->\mathbb{N}$,
$$f(n{)}^2-<!--\; -\; -->L=p^{g(n)}q(n)$$
Where $g:\mathbb{N}\to <!--\; \to \; -->\mathbb{N}$ is an unbounded function and q(n) is coprime to n for all n. Clearly, this is also equivalent to say that the equation
$$x^2\equiv <!--\; \equiv \; -->L(\mathrm{mod}{p}^n)$$
Is soluble for infinitely many n. But we know that if L is a quadratic residue modulo $p^n$, it is also a quadratic residue modulo $p^{n-<!--\; -\; -->1}$. Therefore, the statement resolves to the following proposition:
There is (not) a non-square integer L such that it is a square modulo $p^n$ for every $n\in <!--\; \in \; -->\mathbb{N}$.
Which I know neither is true or not, nor how to start off decently. Though I have not found any of them, and I believe that such L does not exist, i still cannot find out a complete proof. The proposition can indeed be generalized into the following
There is (not) a non-kth power L such that it is congruent to a kth power modulo $p^n$ for all $n\in <!--\; \in \; -->\mathbb{N}$

What is the solution for $y(t)=e^{-<!--\; -\; -->\frac{t}{\mathrm{\tau <!--\; \tau \; -->}y(t)}}$?
A simple quadratic flow model leads to the following apparently simple equation
$$y(t)=e^{-<!--\; -\; -->\frac{t}{\mathrm{\tau <!--\; \tau \; -->}y(t)}}$$
where the flow, y is a function of time, t and τ is a constant.
But is there a closed form solution for y(t) just in terms of t and $\mathrm{\tau <!--\; \tau \; -->}$?
Or can this only be solved by numerical means ?

How do I find the coordinates of symmetric matrices after finding the eigenvalues, eigenvectors and eigenbases?
I've been playing around with Symmetric matrices and orthogonal bases of said Symmetric matrices, but I cannot figure out how to find the coordinates.
So, let's say that I have a quadratic function:
$6x_1^2+4x_1{x}_2+3x_2^2$
Well now I know that this forms a matrix of $A=\left(\begin{array}{cc}6& 2\\ 2& 3\end{array}\right)$
So now I have to find the eigenvalues of this matrix which are ${\mathrm{\lambda <!--\; \lambda \; -->}}_1=7,{\mathrm{\lambda <!--\; \lambda \; -->}}_2=2$.
and now I have two Eigenbases of
$E_7=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left(\left(\begin{array}{c}-<!--\; -\; -->2\\ -<!--\; -\; -->1\end{array}\right)\right),E_2=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left(\left(\begin{array}{c}1\\ -<!--\; -\; -->2\end{array}\right)\right)$
and the orthogonal bases of these two are just the lengths multiplied by the matrices, so
$$\stackrel{\to <!--\; \to \; -->}{w_1}=\frac{1}{\sqrt{5}}\left(\begin{array}{c}-<!--\; -\; -->2\\ -<!--\; -\; -->1\end{array}\right)\stackrel{\to <!--\; \to \; -->}{w_2}=\frac{1}{\sqrt{5}}\left(\begin{array}{c}1\\ -<!--\; -\; -->2\end{array}\right)$$
But the problem I'm having is that I need an equation $q(\stackrel{\to <!--\; \to \; -->}{w1}{c}_1+\stackrel{\to <!--\; \to \; -->}{w2}{c}_2)=c_1^2+c_2^2$
But how do I find these coordinates? I read my book and it wasn't clear on what I'm supposed to be doing to obtain them.

How to prove the existence of a minimum of a quadratic function of two variables?
$$f(x,y)=A{x}^2+2Bxy+C{y}^2+2Dx+2Ey+F,\text{where }A>0\text{ and }{B}^2<AC.$$
Prove that a point (a,b) exists which f has a minimum.
I figured out that there is no stationary point for this equation.
So, Hessian Matrix seems not helpful.
In my book, it says that "change quadratic part to sum of squares
but, Can't think of any way to change it to sum of squares.
Also,
Why $f(a,b)=Da+Eb+F$ is at this minimum..?

Find an equation of the quadratic function with zeros at (0,0) and (6,0) with $f(5)=-<!--\; -\; -->15$
write the equation of the quadratic function with zeros at (0,0) and (6,0) with $f(5)=-<!--\; -\; -->15$
So, I know how to get the equation from the zeros, but I am confused with what I am supposed to do with "$f(5)=-<!--\; -\; -->15$".

Finding the stationary point of a type of hyperbola?
I know that to find stationary points on a function, we need to differentiate the function and set that = 0.
But how can we find the stationary points of the below function?
$$y=\frac{1}{x}+\frac{1}{x^2}-<!--\; -\; -->\frac{1}{x^3}$$
When I differentiate it, I get fractions with x in the denominator... I try to factorize and I get a quadratic equation with non-real roots and then $\frac{1}{x^4}$ which $=0$. But the only way for $\frac{1}{x}=0$ is if $x=\mathrm{\infty <!--\; \infty \; -->}$.

Finding value of k for which $fg(x)=k$ has equal roots?
I've been going through this community and I Find this really helpful. About me(I know I should be precise but ya), I'm just a highschool student who can't afford any coachings/schools. Self schooling being my only option I'm trying to teach myself mathematics from some torrented books. I am on functions and their graphs and stuck with one question
The functions f and g are defined for $x\in <!--\; \in \; -->R$ by $f(x)=4x-2x^2;g(x)=5x+3$.
(i) Find the range of f. (ii) Find the value of the constant k for which the equation $$gf(x)=k$$ has equal roots.
Now, I do understand the composition of functions but I just don't understand what they are asking in this case, $g(fx()=k$ would result in
$$20x-<!--\; -\; -->10x^2+3=k$$
Now this is a quadratic equation of second degree which should have 2 roots/solutions, but that's the case if the right hand side was zero and not k. I have absolutely no idea how to tackle this question and what's being asked in part ii of the question. Anyhelp would be highly appreciated

How can I solve this system of equation?
1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form:
$a{x}^2+bx+c=0$
If the value of the determinant of the quadratic equation is negative (i.e. $\mathrm{△<!--\; △\; -->}<0$), the program should write an appropriate message.
1.10. Modify the 1.9 so that the parameters A and a are functions of the parameter $\mathrm{\varphi <!--\; \varphi \; -->}$, e.g.
$A(\mathrm{\varphi <!--\; \varphi \; -->}=A(1.0+0.25\sin <!--\; \; -->(\mathrm{\varphi <!--\; \varphi \; -->}))$
$a(\mathrm{\varphi <!--\; \varphi \; -->})=a(1.0+0.5\cdot <!--\; \cdot \; -->|\sin <!--\; \; -->(\mathrm{\varphi <!--\; \varphi \; -->})|)$
In the program, declare functions for the above equations and save the results in an array.
I have solved the 1.9 as that was easy. But I am stuck with the following:
$\begin{array}{}\text{(1)}& f(x)=f(1+0.25\sin <!--\; \; -->(x))\end{array}$
$\begin{array}{}\text{(2)}& g(x)=g(1+0.5|\sin <!--\; \; -->(x)|)\end{array}$
My attempt:
Given,
$f(x)=f(1+0.25\sin <!--\; \; -->(x))$
Now,
$f(1+0.25\sin <!--\; \; -->(x))=\frac{a}{16}{\sin }^2<!--\; \; -->x+(\frac{a}{2}+\frac{b}{4})\sin <!--\; \; -->x+(a+b+c)$
Let, $\sin <!--\; \; -->x=y$
So, the above equation becomes: $y=\frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}$
Hence,
$\begin{array}{rl}\sin <!--\; \; -->x& ={\displaystyle \frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}}\\ x& ={\sin }^{-<!--\; -\; -->1}<!--\; \; -->\left({\displaystyle \frac{-<!--\; -\; -->(2a+b)\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}}\right)\end{array}$
Are this procedure and solution correct?

When does this least squares analytical solution based on zeros of partial derivatives start providing more than one solution?
If I want to fit a quadratic function of two variables to some data, I can use
$$f(x,y)=c_1{x}^2+c_{2}xy+c_3{y}^2+c_{4}x+c_{5}y+c_6$$
$$\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}{c}_i}\underset{j}{\sum <!--\; \sum \; -->}{(z_j-<!--\; -\; -->f(x_j,y_j))}^2=0$$
to obtain six equations, and then endeavor to solve them.
I've done it for one variable not two, but I'm guessing the process is straightforward.
If I extend this to more variables, and to higher order than 2, when will the analytical expressions be at risk for having multiple solutions?

Critical Points of Quadratic Forms
Just a question about finding critical points (points where the differential is not surjective). I have the equation
$$f(x)=x^{t}Ax$$
where A is a symmetric n by n matrix and x is an element of ${\mathbb{R}}^n$. Now, I've found the derivative of the function to be
$$df(x)=2x^{t}A=2Ax$$
by symmetry of A. I'm having trouble determining values in ${\mathbb{R}}^n$ where the derivative is not surjective. It's clear that for $x=(0,...,0)$, the derivative is not surjective, but my intuition is that for any other value of x, and for any value in ${\mathbb{R}}^n$, we can choose a corresponding A. Am I thinking about this correctly? Is my derivative correct for the quadratic form $f(x)=x^{t}Ax$?

Find roots for an equation with quadratic, linear and log terms?
I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form
ax2+bx+clogx=0.
considering the natural log. Wolfram alpha is leading me to expressions involving the Lambert W (product log) function when I include either the quadratic term or the linear term (but not both) and analytic approximations when I supply real values for the coefficients.
This is OK, but does a more general solution exist in terms of the coefficients?

Finding a Lyapunov function for modified quadratic form
I am trying to construct a Lyapunov function to show global asymptotic stability for a somewhat difficult system of equations. I was hoping people might have some suggests on what types of equations to try. The system of n equations is given by:
$\stackrel{\dot{}<!--\; \dot{}\; -->}{x}=MD(x)Rx-<!--\; -\; -->g(x)$
where M is a symmetric matrix, $D(\stackrel{\to <!--\; \to \; -->}{x})$ is a diagonal matrix with $\stackrel{\to <!--\; \to \; -->}{x}$ on the diagonals, and g(x) is a positive-definite function that is linear in x. If M is the identity then this reduces to a standard quadratic form. But for M otherwise, I am unable to control this system. You can assume that M and R are full rank, and that M is positive definite.
I have been considering equations of the form:
$V(x)=\underset{i}{\sum <!--\; \sum \; -->}(B{x}^{\ast <!--\; \ast \; -->}{)}_i\log <!--\; \; -->(\frac{(B{x}^{\ast <!--\; \ast \; -->}{)}_i}{(Bx{)}_i})$
such that:
$\stackrel{\dot{}<!--\; \dot{}\; -->}{V}(x)=\underset{i}{\sum <!--\; \sum \; -->}\frac{(B{x}^{\ast <!--\; \ast \; -->}{)}_i}{(Bx{)}_i}(B\stackrel{\dot{}<!--\; \dot{}\; -->}{x}{)}_i$
for some matrix B. For example, if B is the identity matrix, then this gives:
$\stackrel{\dot{}<!--\; \dot{}\; -->}{V}(x)=\underset{i}{\sum <!--\; \sum \; -->}\frac{x_i^{\ast <!--\; \ast \; -->}}{x_i}{\stackrel{\dot{}<!--\; \dot{}\; -->}{x}}_i$
But I am starting to think that this general form is not an appropriate form to work with, as I cannot make any progress. My idea is that B should be related to M and/or R, but no luck.
Are there any obvious things I am missing or alternate Lyapunov functions that I should explore?

Maximum value of a function with 2 variables
Can someone help me finding maximum value of a ratio in quadratic function in 2 variables using proper mathematical methods.?
Question is as below.
If x and y are real numbers such that $x^2-<!--\; -\; -->10x+y^2+16=0$, determine the maximum value of the ratio y/x
I know there is Ramban method to solve this. Taking $y/x=k-<!--\; -\; -->-<!--\; -\; -->>y=kx$ and forming equation in x, then applying ${}^2-<!--\; -\; -->4ac>=0$ for max min value of k.
Is there any way to using differentiation ?

Factoring Polynomial using Quadratic Equation
I'm trying to use the quadratic equation (QE) to factor a degree 2 polynomial into the format: $(x+a{)}^2$, where a is any real number.
This works great for equations like:
(1) $x^2+2x+1$
The QE gives roots $x=\{-<!--\; -\; -->1,-<!--\; -\; -->1\}$, and $(x+1{)}^2=x^2+2x+1$.
This approach fails for other polynomials such as:
(2) $9x^2+36x+36$
The roots found with the QE are $x=\{-<!--\; -\; -->2,-<!--\; -\; -->2\}$. But $(x+2{)}^2\ne <!--\; \ne \; -->9x+36x+36$.
9 is the GCF from the coefficients of (2), so (2) can be rewritten as:
$9\times <!--\; \times \; -->(x^2+4x+4)$
and by using the QE on the second term, this equals:
$9\times <!--\; \times \; -->(x+2{)}^2$. And $9\times <!--\; \times \; -->(x+2{)}^2=9x+36x+36$ (2), so this seems to be a better approach.
So my question is: must the input to QF be a polynomial that has coefficients with $GCF=1$, for the result to give roots that can be used to reconstruct the original function?