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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### 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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### Probability and Sum of All Solutions [closed]

Given x=5.5386388265268286537811327028332 Then 2x^2 - 3x + 1 = 0 or 2*5.5386388265268286537811327028332^2 - 3 * 5.5386388265268286537811327028332x + 1 = 0 The question Im asking, does x hold true for ...
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### Finding solution fby changing ellipses [closed]

Lets say we have two, $2$ dimensional positive integer vectors $X_0$ and $X_1$. We know that their sum is $[P,P]= X0+X1$. Lets also say assume that We get $Z=X1-X0$ from the intersection of $2$ ...
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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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### compose a quadratic equation with integer coefficients [closed]

compose a quadratic equation with integer coefficients having roots $\frac{2-x_1}{x_2}$ and $\frac{2-x_2}{x_1}$, where $x_1$ and $x_2$ are the roots of the equation $3x^2+2x-9=0$
1 vote
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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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### How would you go about graphing a parabola on a graph and determining what the $x$ is [closed]

Example $-x^2-2=0$. How would you determine this $x$ and how to graph it, is there a formula you must follow? *edit this is confusing me very badly all I've seen was the question like that and it ...
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### Quadratic equations with a common root - does the argument work both ways?

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 ...
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### Question regarding max and min of function when derivative cannot be 0

This question is in regards to finding the range of the following function $$x^3 - x^2 + x + 1$$ I decided to find the maxima and minima of this function by computing its derivative, the derivative of ...
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### Quadratics - Sum of areas is minimum...

I am trying to help my daughter with this question: ...
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### Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?

I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself. I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
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### Graphing machine for a quadratic equation representing braking speed of a vehicle over time

I am solving for a particular situation. A two vehicles approach an intersection and are on a collision course. To avoid the collision, one vehicle needs to apply the brakes such that it slows to ...
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### Ruling out finite time explosions to infinity of Riccati differential equations

Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions. Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive ...
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### If $A$ is a symmetric positive definite matrix, show that $f(x) = x^TAx$ is convex.

Let: \begin{gather*} f: \mathbb{R}^n \to \mathbb{R}, \quad A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, x \in \mathbb{R}^n, c \in \mathbb{R} \\ f(x) = x^T A x + b^T x + c \\ \end{gather*} If ...
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### I can't seem to understand the proof of the Cauchy-Schwarz inequality

The proof said to use the quadratic polynomial: $$P(t) = \sum_{i=1}^n(a_it - b_i)^2$$ by which we notice that $P(t) \ge 0$ , but then this is the part which I didn't understand where we conclude that ...
1 vote
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### Prove or give counter example of quadratic inequality

I have two finite probability mass functions (pmfs) $P(x)$ and $Q(x)$ on the same support $(0,1,\ldots,n)$. Let $(p_0,p_1,\ldots,p_n)$ and $(q_0,q_1,\ldots,q_n)$ be the probability vectors from the ...
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1 vote
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### How do you rearrange an equation with the product of floor functions?

Edit: I spent some time trying to generalise this formula for any build plate $l$ and $w$ and have realised a very small error in one of the quadratic equations below which I will correct tomorrow ...
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### Factoring $x^2 + 4x -3$ step by step.

I have the following equation that I need to factor: $x^2 + 4x -3$. I cannot use the factoring by grouping method as there are no integers that add up to $4$ and give $-3$ when multiplied. What method ...
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1 vote
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### How to find factors of a number that add to a certain sum?

For example, when finding the roots of this quadratic equation: $x^{2}+8x-9=0$ I would write $x^2-x+9x-9=0$ Then write that expression as the product of two linear expressions: $(x+9)(x-1)=0$ Then I ...
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From what I understand: $D > 0$ and a perfect square $\Longrightarrow$ Real and Rational Roots $D > 0$ but not a perfect square $\Longrightarrow$ Real and Irrational Roots $D = 0$ $\... • 31 2 votes 1 answer 42 views ### Intersection of parabola and line Suppose that a parabola and line are given by $$y = 2x-k, y = x^2-(k+2)x+2k$$ and if one of the points at which they intersect is on$x$-axis, how can we find the ordinate of the other intersection ... 1 vote 0 answers 79 views ### Multiply two lines together Excuse me , I had a strange question :). In algebraic geometry, a quadratic expression can be written as the product of two different linear equations, for example: y = x² + 3x - 4 can be factored as ... 0 votes 2 answers 48 views ### Proving a quadratic function is bijective in given range Prove that$f: [0,\infty[ \to [-5,\infty[$defined as$f(x) = 4x^2+4x-5$is bijective. I can prove it graphically but not algebraically. 0 votes 0 answers 46 views ### Upper bound of diagonal matrix multiplied with Hermitian matrix Suppose that${\bf R} \in \mathbb{C}^{n \times n}$is a Hermitian matrix, and${\bf D}$is a diagonal matrix with main diagonal being${\bf d} \in \mathbb{C}^{n \times 1}$. I am looking for the ... • 41 2 votes 1 answer 40 views ### Can there be more solutions to following equation? Consider the following equation for$t > 1$: $$(t + \sqrt{t^2 - 1})^{x^2 - 2x} + (t - \sqrt{t^2 - 1})^{x^2 - 2x} = 2t$$ If we let$u = t + \sqrt{t^2 - 1}$then$\frac{1}{u} = t - \sqrt{t^2 - 1}$... • 155 -1 votes 1 answer 88 views ### How can I find the other root of a quadratic equation with 1 root and a unknown b value? [closed] The information I was given was, "If$6x^2 + kx -42 = 0$has one root as$\frac{-7}{6}\$, then what is the other root?
I need to find all numbers $$6x^2 + 2x \equiv 20 \pmod{513}$$ and I'm having a trouble since 513 is not prime number I tried looking for roots but i end up with \begin{...