Questions on quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

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94 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on Fibonacci quadratic residue: $F(x)^2 mod F(y)$ = { if y is even: $(F(|x-y|)^2$ } { if y is odd: $-(F(|x-y|)^2$ } for ...
4
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0answers
73 views

Quadratic system of ODEs

I have a quadratic ODE system that looks like this: $\dot{x}=Ax+diag(x)Nx$ where $x \in R^n$ and $A,N \in R^{n \times n}$ and $diag(x) \in R^{n \times n}$ is a diagonal matrix in which $x$ is its ...
3
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66 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...
3
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0answers
267 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
3
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81 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
2
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0answers
31 views

Second-order Quadratic Constraint

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1^T\theta\\ \text{subject to} & ...
2
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0answers
18 views

Extraction of quadratic terms with state-space representation

I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ . ...
2
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55 views

Does $2^2=4$ imply $2=\pm \sqrt{4}$?

I read the square root property from the book, College Algebra by Raymond A Barnett and Micheal R Ziegler that, The square root property says, If $A^2=C$ then $A=\pm \sqrt{C}$ I took the ...
2
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0answers
48 views

$A(x+p)²-B(x-p)²=y$, historical/math reference

I'm trying to build a reminder of all that I found about the quadratic function over the years. Here I came across this form of quadratic equation that I did not know: A(x+p)²-B(x-p)²=y I have no ...
2
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31 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
2
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147 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
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51 views

Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
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22 views

Reference about quadratic forms with discriminant 1

When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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21 views

Optimization of a Quadratic on a Linear Variety

We have a linear subspace $L_j := L[s_0,s_1,...,s_j]$ and a linear variety: $x_0 + L_j := [x_0 + y : y \in L_j]$ and a standard quadratic cost function $V(x) = a + b^Tx + 0.5x^TCx, \ \ \ C^T = C ...
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48 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
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53 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
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25 views

All solutions to Quadratic matrix polynomials

I am after two things: 1- algorithms for finding all solutions of possibly large quadratic matrix equations of the form $AX^2+BX+C=0$ 2- (if possible) software implementing the algorithms ...
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17 views

Probabilty of even number of games won/lost uses auxiliary variables for a quadratic equation. Why?

In a problem of finding the probability that an even number of games (even S) not being lost in $l$ games, I read the following explanation : "We form the equation, $x^2 - 4rx + 2r^2 = 0$, and ...
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26 views

Linear constraints in Quadratic equation

I have been going through this paper, and wish to implement the same algorithm in java. I have also managed to write equivalent code for the same, but I have not completely understood the mathematics ...
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53 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
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0answers
48 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...
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81 views

Showing DO NOT exist GCD of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$.

Showing DO NOT exist gcd of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$. I tried it. Suppose $d$ is GCD of $6$ and $2+2 \sqrt(-5)$. then there exist $x,y \in \Bbb ...
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45 views

Interchanging from exponential form to log form

Shouldn't the answer be x = loge(everything else in the bracket) why is the loge function divided by "k" ???
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21 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
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21 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
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65 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
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0answers
83 views

Quadric equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $f(x)=ax^{2}+bx+c$ in $\mathbb{Z}/n\mathbb{Z}$, we know that this quadric equation $f(x)=0$ has exactly 8 ...
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88 views

Can the following quadratic equation be solved for M without iterating over possible values

I have developed this equation for a piece of software I am writing. Not being very mathematically minded I am stumped at how I can solve for M without iterating over all possible values for M. The ...
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41 views

Calculating $(x_1^{x_2})(x_2^{x_1})$ in a quadratic equation

Supposing $S=x_1+x_2$ and $P=x_1x_2$ where $x_1$ and $x_2$ are the roots of the quadratic equation $ax^2+bx+c=0$ (it's clear the equation is equivalent to $x^2-Sx+P=0$ where $S=-b/a$ and $P=c/a$), how ...
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0answers
70 views

Having trouble with finding a Quadratic Expression

I am having trouble trying to work out a quadratic expression for uni, being a external student its hard to find help. My problem is Perimeter = 1000m Part 1 Solve L in terms of w in regard to the ...
0
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11 views

Second-order quadratic model with bias term

I have 3 points in the 3-d space and I would like to estimate the parameters of a second-order quadratic model with a bias term $z=f(x,y)=ax^2+bxy+cy^2+dx+ey=\theta^TQ\theta+\eta^T\theta$ where the ...
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51 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
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21 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...
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10 views

Solution to Equation involving Volatility

The following question will have little context, though, it is not relevant. To summarise though, I am trying to find solutions $u$ and $d$ to the following equation given that $d = \frac{1}{u}$: ...
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26 views

Line intersection with Sphere

I'm trying to get a formula for calculating intersection points of a line with a sphere (3d space). I've been following this one: Wiki Line-sphere intersection But I'm 99% sure that this one is ...
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34 views

The property of roots of quadratic equation

I have a problem with two tasks: Given a quadratic equation $ax ^ 2 + bx + c = 0 $ (roots can be complex or real), $a, b, c \in Q$. Prove that ${x_1} ^ m + {x_2} ^ n \in Q$. We have a trinomial ...
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13 views

Alternative method for y-vertex calculation

So, I've been wondering the following: If you can determine the x coordinate of the vertex of a quadratic function by averaging the x coordinates of both roots, would it be possible to determine the ...
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14 views

Determine the multi-dimensional relationship given the data

I have a dependent variable - A and 3 independent variables, H,V and N I have a data for all the variables and dependency relationship is based on my operational knowledge. I'd like to know what ...
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13 views

Constrained Motion Study

I'm working on a motion study for a disk moving within a mechanical enclosure and I'm having trouble reducing my equations. The system can be defined as 4 circles which are bound inside each other. ...
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0answers
21 views

Quadratic Polynomial Formula [$ax^2+b_1x+c$ vs. $ax^2+2b_2x+c$]

I thought the quadratic formula was $ax^2+b_1x+c$. However, in my linear algebra book when they deal with $x^TKx$, they use the formula $ax^2+2b_2x+c$. I understand that $b_1$ = $2b_2$, but what is ...
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27 views

Calculating speed of an object between time intervals using quadratic function

I'm actually getting stuck with a quite difficult part of a 1D math problem using quadratic functions. An object is propelled with an initial speed, has its altitude given after t seconds by the ...
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31 views

area of a sprinkeler.

A rectangular lawn has an area of 677 square meters. Surrounding the lawn is a flower border 4 meters wide. The border alone has an area of 548 square meters. A circular sprinkler is installed in the ...
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21 views

Numerical method to find coefficient of quadratic function given target skewness

I have two samples, $X$ and $Y$, and for both I calculate the sample skewness Sk$(X)$ and Sk$(Y)$. My objective is to find $d$ such that, given $Z = X + dX^2$, Sk$(Z) = $ Sk$(Y)$. The coefficient of ...
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41 views

How to show that $2^x$ is not in $O(x^2)$?

This is from Discrete Mathematics and its Applications I am working on 2e. I knew right off the bat from previous computer science courses that 2^x is not in O(x^2). I am having a difficult time ...
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29 views

Why can't this inequality hold true for all n > k?

This is from Discrete Mathematics and its Applications I am having trouble with why the "inequality n <= C cannot hold for all n with n >k". Is this reasoning for this that there is no largest ...
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15 views

Is there an efficent way to solve large systems of purely quadratic equations?

I have the following system of quadratic equations $$ b_1 = \sum_{k=1}^R x_{i_1, k} \ y_{j_1, k} $$ $$ \vdots $$ $$ b_p = \sum_{k=1}^R x_{i_p, k} \ y_{j_p, k} $$ where $i_1, \ldots, i_p \in ...
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37 views

quadratic formula for polynomials with variable coefficients

I have trouble calculating equations like the one in last comment in the first answer; Solve system of 3 equations there are variable coefficients which I can calculate using quadratic formula - if ...
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28 views

Convert Bearing Degrees to Slope

I'm working on a mapping application that sketches a graphic on the map. The user will input the distance of a line segment and provide a bearing for the line segment. For example, a line segment is ...
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0answers
23 views

How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular ...
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24 views

Find two integers $a, b$ for given integer $c$, so that $c=a^2\pm b^2$

Given a positive integer $c$: Find two other positive integers $a$ and $b$, so that $c=a^2 + b^2$ and/or $c=a^2 - b^2$. I've already got a solution for any odd $c$: $c = (x+1)^2 - x^2 = 2x + 1$ so ...