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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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 $$ay^... 0answers 217 views ### Question: How to find the smallest value x satisfying the equation: x^2 = a \pmod c (known is a and c, c is not the prime)? Question: How to find the smallest value x satisfying the equation: x^2 = a \pmod c (known is a and c, c is not the prime)? Using the Tonelli-Shanks algorithm and the Chinese remainder ... 0answers 134 views ### Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial ax^2+bx+c has real roots if the parameters a,b,c are exponentially distributed ... 0answers 184 views ### Fibonacci Quadratic Residue After some research I have came up with a conjecture on Fibonacci quadratic residue: F(x)^2 mod F(y) = { if y is even: (F(|x-y|)^2 } { if y is odd: -(F(|x-y|)^2 } for values ... 0answers 55 views ### Help me see the connection between exponential functions and quadratic curves We know that the degree 2 equations x^2 + y^2 =1 and x^2 - y^2 =1 can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it ... 0answers 89 views ### Why do perfect square values to ax^2 +ax +1 form an exponential function? While playing around with numbers using Python, I found that the set of values of x which fulfilled$$ax^2 + ax +1 = p^2$$Where p is an integer form an exponential function. For example,$$3x^2 + ...
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Do you know a way to model what I would call a non-symmetric quadric surface? see the pictures : ,
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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For a function: $$f(x)=\frac{(1-x^2)}{(1+x^2)-2 x \cos \omega}$$ Let $x=-\frac{1}{3}$ That means that $$f(-\frac{1}{3})=\frac{(\frac{8}{9})}{(\frac{10}{9})+\frac{2}{3} \cos \omega}=\frac{4}{5+3 \... 0answers 125 views ### Can a quadratic approximation be calculated with a table calculator? The continued-fraction-method allows to calculate a linear approximation of a real number with a table calculator. But I do not know an analogue method for a quadratic approximation. Given a real ... 0answers 123 views ### Suppose that a function f(x)=ax^2+bx+c, where a,b,c are real constants, satisfy the relation.. Suppose that a function f(x)=ax^2+bx+c, where a,b,c are real constants, satisfy the relation$$-1\leq f(x)\leq 1 $$for all -1\leq x\leq 1, then the maximum value of f'(x) is I think the ... 0answers 136 views ### What is relation between a particular root of two polynomials? We have$$x^3+(m+n+p-1)x^2-((m+n)(1-p)+2p-1-mn)x-(p-1)(m-1)(n-1)=0$$in which m,n\ge2, p\ge1 are natural numbers. All the three roots of this cubic are positive. Let \lambda be the least of them. ... 0answers 71 views ### Help solving the quadratic equation ax^2-4bx+4bc-\frac{d^2}{a}=0 I have been struggling to solve this quadratic equation in the variable x with integral coefficients:$$ax^2-4bx+4bc-\frac{d^2}{a}=0$$a\neq 0 of course.How do I ensure that x is an integer? ... 0answers 105 views ### Problem on Bivariate normal distribution Let X_1 and X_2 have a bivariate normal distribution with parameters \mu_1 = \mu_2 = 0 and \sigma_1 = \sigma_2 = 1 and \rho = 1/2 Find the probability that all the roots of X_1x^2+ 2X_2x + ... 0answers 72 views ### Solution to the following set of equations Is the solution to the following:$$a^2+b^2=1c^2+d^2=1ad+bc=1$$still a=d=\cos z, c=-b=\sin z, when a,b,c,d \in \mathbb C? 0answers 141 views ### Why the quadrature formula is exact one not an approximation? I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ... 0answers 147 views ### Taylor expansion need help understanding. I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ... 0answers 180 views ### Lagrange multiplier expression I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & \theta^... 0answers 272 views ### Extraction of quadratic terms with state-space representation I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: T_i + f_i(\omega, \alpha) = 0;\;i:1-4 . ... 0answers 61 views ### A(x+p)²-B(x-p)²=y, historical/math reference I'm trying to build a reminder of all that I found about the quadratic function over the years. Here I came across this form of quadratic equation that I did not know: A(x+p)²-B(x-p)²=y I have no ... 0answers 140 views ### How surfaces intersect in projective spaces Consider this parametrization$$\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)$$Let \mathcal{C} be the image of \phi. I've proved that \... 0answers 29 views ### Multidimensional Quartic Equations I know for the quadratic case (with A an operator):$$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dxDoes any such analogy exist with ax^4 type functions? Either in the finite or infinite ... 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 + ... 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 ... 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}{... 0answers 61 views ### Parametrisation of a general quadratic surface? Consider a general quadratic surface in implicit form: $$ax^2 + by^2 + cz^2 + 2exy + 2fyz + 2gxz + 2lx + 2my + 2nz + d = 0$$ I can parametrise this in the form f(x, y, z(... 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}$$... 0answers 45 views ### What does changing the coefficients of polynomials graph? Quadratic Function Let f(x)=ax^2+bx+c It's obvious that when we change absolute term the vertex of the parabola graphs a vertical line with equation x=-b/2a When we change b, the vertex ... 0answers 90 views ### On a statement in The Joy of Factoring By Samuel S. Wagstaff (Jr.)? In 'The Joy of Factoring By Samuel S. Wagstaff (Jr.)' on page 32 it is mentioned that Euler, Legendre, Gauss, and Chebyshev have observed that in$$ax^2+bxy+cy^2=N$$for two different (x,y) pairs ... 0answers 39 views ### Error in center-focus length of an ellipse I have an equation of ellipse in the form:$$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 ...
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 ...