Articles of quadratics

Relation between quadratic inverse and it's roots

Finding the inverse of a function and finding the roots appear to be a similar procedure. Note the graph below. I took the inverse equation and simply removed the x variable and changed y= to x= and got the following: I’m basically just wondering if this is correct, cause it seems like waste of time […]

High School Projectile Motion and Quadratics

High school students are learning about the basics of solving quadratics and trigonometric ratios, including trigonometric inverses. The eventual goal of their project is to be able to show a reasonable firing solution, given in initial angle $\theta$ and initial velocity $v_0$. Projectile motion is given by $$y=\left(\tan{\theta}\right)x-\left(\frac{g}{2v_0^2\cos^2{\theta}}\right)x^2$$ where $x$ and $y$ are horizontal and […]

Convert quadratic bezier curve to parabola

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ function) that this Bézier curve is a part of? (algebraically) So in the image, the red curve is known, how […]

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 – z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 – (z)^2 + 1) = 0$$ $$z = \frac{1 \pm \sqrt{-3}}{2}$$ But this is incorrect. How to do this […]

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would you solve any set of equations of the form: \begin{matrix} { ax }^{ 2 }+bxy+c=0 \\ d{ y }^{ 2 }+exy+f=0 \end{matrix} where $a, b, […]

Show $p$ prime s.t. $p \not\equiv 1 \mod 3$ is represented by the binary quadratic equation.

I am working on the following question: Let $p>3$ be a prime such that $p \not\equiv 1 \mod 3$. Show that $p$ is not represented by the binary quadratic equation $f(x, y) = x^2 + xy + y^2$. I would appreciate any help.

If $ -3\left(x-\lfloor x \rfloor \right)^2+2(x-\lfloor x \rfloor )+a^2=0$ has no integral solution, then $a$ is

if $a$ is real number and $\displaystyle -3\left(x-\lfloor x \rfloor \right)^2+2(x-\lfloor x \rfloor )+a^2=0$.has no real integral solution, Then all possible values of $a$ lie in the interval.. (where $\lfloor x \rfloor $ means floor function of $x$) $\bf{My\; Solution::}$ Let $x-\lfloor x \rfloor = \{x\}= y$, where $ 0 \leq \{x\}<1\Rightarrow 0\leq y<1$. Then […]

How to factor the polynomial $5x^2 – 2x – 10$?

I want to factor this polynomial without using the quadratic formula. So, as the first step I multiply coefficient of $x^2$ with $-10$. The product obtained is $-50$. However, there are no two factors of $-50$ that add up to $-2$ and give the product as $-50$. So, it becomes evident that the polynomial would […]

Cyclical System of Quadratic Equations with Four Unknowns

Solve this system of equations in ℝ (k ∈ [0,1]): $\ k-x^2=\ y$ $\ k-y^2=\ z$ $\ k-z^2=\ u$ $\ k-u^2=\ x$

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x – 9)^2 + (y – 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that the first circle’s centre is at $(9,5)$ and has a radius of […]