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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### can anyone tell me what this symbol imply [closed]

It has greater than symbol with less than symbol Image link
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### Solve $x^2+6x-15120=0$

Solve $x^2+6x-15120=0$ So I could just use the quadratic equation and get the answer. However, I have factored $15120 = 2^4 \cdot 3^3 \cdot 5 \cdot 7$. I know I need two factors with a difference ...
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Finding all real values of $k$ so that $$f(x)=(x^2-x+1)^2-2kx(x^2+x+1)+(x^2+x+1)^2=0$$ has two or four real roots. We can re-write this equation as $g(y)=y^2+ky+1-k=0$, where $y=\frac{x^2-x+1}{x^2+x+... • 43.4k -2 votes 1 answer 52 views ### Let$\alpha, \beta$be the roots of$x^2-a_1x+1=0$, and consider the sequence of numbers$a_r(r\ge0)$with$a_0=1$and$a^2_{r+1}=1+a_r.a_{r+2}$Let$\alpha, \beta$are roots of equation$x^2-a_1 x+1=0$and consider sequence of numbers$a_r,\;r\geq0\;$with$a_0=1\;$and$a_{r+1}^2=1+a_r\cdot a_{r+2}.\;Then which of the following is/are true?... • 8,310 0 votes 0 answers 52 views ### Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ... 0 votes 0 answers 17 views ### Question about 2D-Fresnel integration Thank you for reading my question! I am trying to get the following integration: \begin{align*} \int_0^\infty\int_{-\infty}^{+\infty}e^{j(a(x-x_0)^2+b(y-y_0)^2+c(x-x_0)(y-y_0))}dxdy \end{align*} ... • 335 0 votes 1 answer 30 views ### Quadratic where roots and coefficients together form Arithmetic Progression Background I was reading this post: A.P. terms in a Quadratic equation. And wondered the following: Given a quadraticax^2+bx+c=0$which has roots$x=m,x=n$, is it possible for$a,m,b,n,c$to be ... • 2,762 1 vote 0 answers 88 views ### How to find x if its in the denominator and on the other side of equal sign. [duplicate] $$\frac{a}{b+xc} = x$$ So I have$x$in the denominator on the left side and also$x$on the right.$a$,$b$, and$c$are known and I need to find out for$x$. How should I go about it? Edit: Thanks ... • 11 0 votes 0 answers 102 views ### Ages of Xander, Alice, Bob, and Carol I need help with quadratics and derivatives for this problem: Problem: Xander, Alice, Bob, and Carol all have unknown ages. We are told that if we multiply the age of Alice by the square of the age of ... • 24.9k 0 votes 1 answer 21 views ### Find the canonical form of each equation and classify the quadratic curve I am trying to get this equation to the canonical form using Lagrange Method (completing the squares). $$5x^2 + 4xy + 8y^2 - 32x - 56y + 80= 0$$ The first step is to get rid of the$4xy$term, which ... 4 votes 2 answers 472 views ### Quadratic with integer roots This question is from the 1991 Russian Mathematics Olympiad Grade 10 Final round. I have searched as best I can on this site and AoPS for a solution to compare to my own but with no luck. The question ... • 2,762 -2 votes 1 answer 83 views ### Why do we need this inequality? [closed] I have been going through a state maths exam and am not able to answer the following question. I understand that$k>-1/4$. However, in the solutions it is also stated that$e^{-x}>0$. Why does ... 3 votes 0 answers 64 views ### Simplifying a quadratic expression under square root I am trying to simplify the following expression $$\sqrt{R}:=\sqrt{a^2(u_1-u_2)^2+b^2(u_1+u_2)^2-2 ab (u_1^2+u_2^2-2)}$$ I have been staring at it for a while in the hopes of getting rid of the square ... • 81 1 vote 1 answer 41 views ### Quadratic equation with a relation between its coefficients Given the quadratic equation$ax^2+bx+c=0$, where$a, b, c \in\mathbb{R}$such that$4(a+b)+7c=0$,$(a\neq0)$prove that: All of the quadratic's roots are real. Atleast one of the roots is in the$[0,...
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I'm struggling to follow the derivation in the steps below, which are copied from a textbook (Agarwal, Foundations of Analog and Digital Circuits). I can't follow the derivation, which is solving (14....
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### Is it possible for a quadratic function to not have $y$-intercept?

I know quadratic functions may have up to two $x$-intercepts, but can they have no $y$-intercepts? Edit: I mean quadratic functions which output parabolas (i.e. $f(x)=ax^2+bx+c$ where $a\neq0$). So ...
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### prove that the point where the gradient of a quadratic function is equal to 0 is always a maximum or minimum point

For context, I am an a level maths student going back through all the content on the course and trying to understand everything that was taught throughout the year with proofs or at the very least, a ...
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### What is the difference between roots and zeroes? [duplicate]

Suppose I have a polynomial of degree 6. It crosses the x-axis at 3 distinct points, and the graph of the polynomial touches the x-axis at one of those 3 points (a repeated root). Question 1: What is ...
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### Non-Euclidean projections

Background Let $y\in\mathbb{R}^2$ be a given point and let $V_1,V_2\in\mathbb{R^2}$ be the vertices of a given segment. Define the projection of $y$ over the segment $s=(V_1,V_2)$ as \begin{equation*} ...
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### Question regarding sign of a trignometric quadratic functions

This question is regards of the following problem If $\cos^2(x) + (1-c)\cos(x) + 2c - 6 \geq 0$ for every $x \in R$ than the true sets of values of $c$ is, I tried to solve the above problem as ...
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### Solution of the quadratic equation including floor function. [closed]

Let $f(x)=x^2 + (x - \lfloor x\rfloor)*\lfloor x\rfloor - 5$. I wonder that how to solve the equation $f(x)=0$. Is there a theorem that tells whether the equation has a solution? Thanks.
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### Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has at least one root in $[0,3)$ - how to show this?

Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has (A) no root in $[0,3)$ (B) all root in $[0,3)$ (C) exactly one root in $[0,3)$ (D) at least one root in $[0,3)$ In this question I ...
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### Cube root equation [closed]

I'm trying to solve: $$\sqrt[3]{x^2} - \sqrt[3]{x} -6 = 0$$ I’ve tried putting the $-6$ on the other side and cubing both sides but no joy at finding value $x$.
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### Ordinate and Abscissa of line parallel to tangent of parabola

A parabola is defined by a focus, $F=(p,q)$, and a directrix, $y=l$ (as shown in the diagram). I want to identify the geometric representation of both an ordinate to the diameter, $x=u$, as well as ...
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### Find constants $p$, $q$, and $r$ for $\frac{16x+1}{px+1} > x+4$ where solution set is $x < q$ or $r < x < 3$ [closed]

Find constants $p$, $q$, and $r$ for $$\frac{16x+1}{px+1} > x+4$$ where solution set is $x < q$ or $r < x < 3$. Attempted to rearrange for quadratic, but resulted in range of values ...
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### GCSE maths level - Is it normal for there to be two possibilities of factorisation for quadratics with a coefficient of x?

The problem and my working It would be useful to know, and its impossible to find anywhere easily on the internet. There also definitely could be a hole in my reasoning too. It was for factorising a ...
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There are no solutions to $x^2+x+3\leq 0$ in $\mathbb{R}$. So why does the solution of $x^2+x+3>0$ mean $x$ can be any $x\in \mathbb{R}$? Can someone please help me by solving the given ...
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### Find the set of values of $\alpha$ so that $f(x)=\dfrac{\alpha x^2+6x-8}{\alpha+6x-8x^2}$ is one one.

Let $f$ be a function defined in its domain given by $f(x)=\dfrac{\alpha x^2+6x-8}{\alpha+6x-8x^2}$. Find the set of values of $\alpha$ so that $f(x)$ is one-one. My attempt As $f(x)$ have to be one-...
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### Determine the pairs $(x,y)$ of integers satisfying $2x^2-3xy+y+1=0$.

the question Determine the pairs $(x,y)$ of integers with the propriety that $$2x^2-3xy+y+1=0$$ my idea I tried writing it as a product of terms but got to nothing useful. Then I applied the quadric ...
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### Estimating the parameters of an ellipse (part 3)

This post is a follow up of this and this previous ones. I've found an explanation for the following formulas \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
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### Show that $\frac{\text{quadratic}}{\text{quadratic}}$ with no common factors is many-to-one

Let $${f(x)=\frac{ax^2+bx+c}{dx^2+ex+f}}$$ hence, $${f'(x)= \frac{(2ax+b)(dx^2+ex+f)-(2dx+e)(ax^2+bx+c)}{(dx^2+ex+f)^2}}.$$ If $f$ is not a continuously decreasing or increasing function then it is ...
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### Estimating the parameters of an ellipse (part 2)

This post is a follow up of this previous one. I would like to clarify why the angle estimator works and how to estimate the axes length. Unfortunately, I still have some trouble with this problem. I ...
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### Sample uniformly on ellipsoid by transforming samples on sphere

Problem Statement Suppose $\pi(x) = \mathcal{N}(0_d, \Sigma)$ is a multivariate normal distribution centered at the origin with covariance matrix $\Sigma$. Given a suitable value $c > 0$, I want to ...
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Here is a question from the Cambridge University 1st year examination from 1889: Prove that if $a+b+c=0$ then each pair of the equations $x^{2}+ax+bc=0$, $x^{2}+bx+ac=0$ and $x^{2}+cx+ab=0$ will have ...