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Questions tagged [quadratics]

Questions about 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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269 views

Is a finite number of quadratic equations in two variables sufficient to solve for the two variables?

Let's say I’m trying to solve a Diophantine problem in two positive integers, $y$ and $q$. Furthermore, let’s say I can derive an extremely large (read: arbitrary) number of equations of the form $$ay^...
5
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0answers
217 views

Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Question: How to find the smallest value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)? Using the Tonelli-Shanks algorithm and the Chinese remainder ...
5
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0answers
134 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
5
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184 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 values ...
4
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0answers
55 views

Help me see the connection between exponential functions and quadratic curves

We know that the degree 2 equations $x^2 + y^2 =1$ and $x^2 - y^2 =1$ can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it ...
4
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0answers
89 views

Why do perfect square values to $ax^2 +ax +1$ form an exponential function?

While playing around with numbers using Python, I found that the set of values of x which fulfilled $$ax^2 + ax +1 = p^2$$ Where p is an integer form an exponential function. For example, $$3x^2 + ...
4
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39 views

Modeling non symmetric quadric

Do you know a way to model what I would call a non-symmetric quadric surface? see the pictures : ,
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2k views

The equations $x^3+5x^2+px+q=0$and $x^3+7x^2+px+r=0$ have 2 common roots, then find the third root of both equations

The equations $x^3+5x^2+px+q=0$and $x^3+7x^2+px+r=0$ have 2 common roots, then find the third root of both equations From the first equation we can say, $\alpha\beta+\beta\gamma+\gamma\alpha=p/...
4
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0answers
136 views

Almost-Linear Sequence of Positive Integers Excluding a Near-Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set (which is a duplicate of $m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+...
4
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0answers
51 views

Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or do ...
4
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622 views

Quadratic equation too hard

I am trying to solve a quadratic equation very hard. Is there any other way to solve this without quadratic formula? $$ x^2(-BE(F+C)^2(G+C)(A+C))))+x(C(F+C))\left [ EBA(D-H)-(G+C)(A+C)(B(D-H)+D(F+C)) ...
4
votes
0answers
169 views

How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
4
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0answers
208 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 ...
4
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0answers
125 views

Quadratic 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 quadratic equation $\,f(x)=0\,$ has ...
3
votes
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83 views

Roots of $ax^2+bx+1 = 0$ are given, then find roots of $x^2+(b^3-3ab)x + a^3 = 0$

Roots of $ax^2+bx+1 = 0$ are given as $\dfrac{1}{\sqrt\alpha}, \dfrac{1}{\sqrt\beta}$. Find roots of $x^2+(b^3-3ab)x + a^3 = 0$ Method 1: I assumed $(\alpha)^{-1/2} =x$ and the new variable in which ...
3
votes
0answers
318 views

Expectation of double quadratic form

I'm reading the 2012-version “The Matrix Cookbook”. On Page 43 Section 8.2.4 “Mean of Quartic Forms” there is a formula that really confuses me: $E[x^TAxx^TBx]=Tr[A\Sigma(B+B^T)\Sigma]+m^T(A+A^T)\...
3
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273 views

Distinct roots of the equation $ax^2-bx+c=0$ in $(1,2)$, where $a,b,c\in\Bbb{N}$

Problem Statement:- If the equation $ax^2-bx+c=0$ has two distinct real roots b/w $1$ and $2$ where $a,b,c\in\Bbb{N}$, then show that $a\ge5$ and $b\ge11$. Attempt at a solution:- As the roots ...
3
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203 views

Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$

Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$ Sum of roots$=\alpha+\beta=\frac{5p+1}{4}$ Given $\beta=1+\...
3
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0answers
137 views

Parabola Terminology

In Danish we call the two halves of a parabola that goes out to each side from the vertex branches like branches on a tree. Is there a name for them in English? Are they just called halves or maybe ...
3
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93 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 (Q^B_{...
2
votes
0answers
55 views

Difficult quadratic equation

Let $$S = \sqrt{x^2+c^2+2x\left(\frac{y}{z}(c+d)-d\right)}$$ and $$B= \frac{S-x+d}{\frac{y}{z}{(c+d)}}, $$ we are supposed to find $d$ (which is independent of x) s.t.: $$ B^2y+B\left(x\left(1-\frac{...
2
votes
0answers
32 views

Is it possible to find a solution to the following $(x(b+x) + c - f) \mod (2x+2) = 0$

Given the equation $$ (x(b+x) + c - f) \mod (2x+2) = 0 $$ Is it possible and if so what is the quickest way to find appropriate value of $x$? Where $x > 0$ The above equation is derived from the ...
2
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0answers
55 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
2
votes
0answers
147 views

necessary and sufficient condition for magnitude of roots of a quadratic equation less than 1 with complex coefficient

I have a general quadratic equation with complex coefficient $$ax^2+bx+c=0$$ where a, b c are all complex numbers. I wonder is there a necessary and sufficient condition to guarantee that all ...
2
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0answers
98 views

Are there infinite many positive integers $n$ such that $n^2 + n +1$ is prime?

I've heard that linear polynomials with proper integer coefficients has infinite many positive integers $n$ such that $f(n)$ is prime, by Dirichlet's theorem. But is there something done with ...
2
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0answers
97 views

How to solve least square quadratic problem using FFT

Let A, B, C and D be some matrix and x be a vector, I want to solve the following optimization problem: $$ \min_x \| Ax -B\|^2_2 + \| Cx-D\|^2_2 $$ my solution: $$ J = \| Ax -B\|^2_2 + \| Cx-D\|^2_2 ...
2
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0answers
59 views

Manifold of Quadric Surface Coefficients

Being a non-mathematician, excuses if I make errors in formulating this question. Consider a general quadric surface of the implicit form: $$ f(x,y,z)=Ax^2+ By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + ...
2
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0answers
91 views

Basic Algebra: How to go backwards?

For a function: $$f(x)=\frac{(1-x^2)}{(1+x^2)-2 x \cos \omega}$$ Let $x=-\frac{1}{3}$ That means that $$f(-\frac{1}{3})=\frac{(\frac{8}{9})}{(\frac{10}{9})+\frac{2}{3} \cos \omega}=\frac{4}{5+3 \...
2
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125 views

Can a quadratic approximation be calculated with a table calculator?

The continued-fraction-method allows to calculate a linear approximation of a real number with a table calculator. But I do not know an analogue method for a quadratic approximation. Given a real ...
2
votes
0answers
123 views

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation..

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation $$-1\leq f(x)\leq 1 $$ for all $-1\leq x\leq 1$, then the maximum value of $f'(x)$ is I think the ...
2
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0answers
136 views

What is relation between a particular root of two polynomials?

We have $$x^3+(m+n+p-1)x^2-((m+n)(1-p)+2p-1-mn)x-(p-1)(m-1)(n-1)=0$$ in which $m,n\ge2, p\ge1$ are natural numbers. All the three roots of this cubic are positive. Let $\lambda$ be the least of them. ...
2
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0answers
71 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
2
votes
0answers
105 views

Problem on Bivariate normal distribution

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + ...
2
votes
0answers
72 views

Solution to the following set of equations

Is the solution to the following: $$a^2+b^2=1$$ $$c^2+d^2=1$$ $$ad+bc=1$$ still $a=d=\cos z$, $c=-b=\sin z$, when $a,b,c,d \in \mathbb C$?
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0answers
141 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
2
votes
0answers
147 views

Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
2
votes
0answers
180 views

Lagrange multiplier expression

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\\ \text{subject to} & \theta^...
2
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0answers
272 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
votes
0answers
61 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
votes
0answers
140 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 $\...
1
vote
0answers
29 views

Multidimensional Quartic Equations

I know for the quadratic case (with $A$ an operator): $$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dx$$ Does any such analogy exist with $ax^4$ type functions? Either in the finite or infinite ...
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0answers
57 views

Solve set of nonlinear equation

I am looking for solving this set of nonlinear equation. \begin{aligned} 2q_1q_3 - 2q_2q_4&=a_1\\ -2q_1q_2 - 2q_3q_4&=a_2\\ - q_1^2 + q_2^2 + q_3^2 - q_4^2&=a3\\ q_1^2 + q_2^2 + q_3^2 + ...
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vote
0answers
51 views

How does this equation yield into the following

$\frac{(a^2 + b^2)} {(ab + 1)} = k$ becomes $x^2-kb \cdot x + (b^2 - k) = 0$ in an example I am attempting to understand, can you please clarify how the quadratic equation follows from the ...
1
vote
0answers
63 views

Rectilinear generatrices parallel to a given plane

Find the rectilinear generatrices of the hyperbolic paraboloid $ P_{h}: \frac{x^{2}}{p}-\frac{y^{2}}{q}=2z $, where $ p,q > 0 $, which are parallel to the plane $ (\pi ):\frac{x}{\sqrt{p}}+\frac{y}{...
1
vote
0answers
61 views

Parametrisation of a general quadratic surface?

Consider a general quadratic surface in implicit form: \begin{equation} ax^2 + by^2 + cz^2 + 2exy + 2fyz + 2gxz + 2lx + 2my + 2nz + d = 0 \end{equation} I can parametrise this in the form $f(x, y, z(...
1
vote
0answers
36 views

Perimeter and area of a rectangle is given.How to find the ratio between two sides of rectangle?

If the perimeter of a rectangle is $160$ meters and its area is $1200$ square meter, then one of its sides must be __________ the other side. My approach to this question is: $$x=\text{one side}$$ ...
1
vote
0answers
45 views

What does changing the coefficients of polynomials graph?

Quadratic Function Let $f(x)=ax^2+bx+c$ It's obvious that when we change absolute term the vertex of the parabola graphs a vertical line with equation $x=-b/2a$ When we change $b$, the vertex ...
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vote
0answers
90 views

On a statement in The Joy of Factoring By Samuel S. Wagstaff (Jr.)?

In 'The Joy of Factoring By Samuel S. Wagstaff (Jr.)' on page $32$ it is mentioned that Euler, Legendre, Gauss, and Chebyshev have observed that in $$ax^2+bxy+cy^2=N$$ for two different $(x,y)$ pairs ...
1
vote
0answers
39 views

Error in center-focus length of an ellipse

I have an equation of ellipse in the form: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$$ I would to calculate the semi-axis major and minor and the center coordinates of the ellipse. I found these ...
1
vote
0answers
111 views

Solving quadratic matrix equations that include a diag operator

Let $\Omega$ and $\Sigma$ be symmetric, PSD matrices, $\lambda$ be a positive scalar, and $I$ be the identity matrix. Further, let the $\text{diag}$ operator set all non-diagonal elements of a square ...