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 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 […]

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 […]

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 […]

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, […]

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 $a$ 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 […]

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 […]

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

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 […]

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