Questions tagged [quadratics]
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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The quadratic equation $x^2 - (c+3)x + 9 = 0$ has real roots $x_1$ and $x_2$. If $x_1 < -2$ and $x_2 < -2$, find value of $c$.
The quadratic equation $x^2 - (c+3)x + 9 = 0$ has real roots $x_1$ and $x_2$. If
$x_1 < -2$ and $x_2 < -2$, then the value of $c$ is ...
I try:
Since there are two real root then
\begin{align}
...
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Why does the leading coefficient of a quadratic trinomial resemble some sort of a slope?
It's known that a unique parabola of the form $y=ax^{2}+bx+c$ exists for any three distinct points, provided that the points are non-collinear and their $x$ coordinates are distinct.
Consider the ...
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How to solve system of 2 arbitrary bivariate quadratic equations over finite field?
I'm in the process of needing a solver for bivariate quadratic system of 2 equations over finite field - this is to estimate the time complexity of breaking an algorithm that I'm designing.
Most ...
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What does a 3D plot of a quadratic look like if the z-axis represents imaginary input values?
I recently saw this video about factoring any quadratic expression without guessing and checking.
The implication, that I probably should have got sooner, is that $f(x) = 0$ then always have a ...
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The real and distinct roots $α$ and $β$ of $x^2+1=\frac{x}{a}$ satisfy $\left|\alpha^2-\beta^2 \right|>\frac{1}{a}.$ What is the domain of $a$?
Problem
The real and distinct roots $α$ and $β$ of $x^2+1=\frac{x}{a}$ satisfy $\left|\alpha^2-\beta^2 \right|>\frac{1}{a}.$ What is the domain of $a$?
I started by converting it into a quadratic ...
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Explain Alpern's method for solving a quadratic modular equation $ax^2 + bx + c ≡ 0 \pmod n$ [closed]
This question will be about Dario Alpern's calculator for $ax^2 + bx + c ≡ 0 \pmod n$.
(Implemented in the function quadmod.c).
I have tried to understand it using ChatGPT.
First I have asked to ...
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Prove that $\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1$ for real numbers
How do we prove that for all $a, b, c \in \mathbb{R}$,
$$\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1.$$
I haven't really made much progress in finding a way to tackle this....
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Rational pre-images of a quadratic polynomial
Suppose I'm given rational numbers $\{y_1,\dots,y_m\} \subset \mathbb{Q}$. It is easy to construct a quadratic polynomial $f(x) = ax^2+bx+c$ such that all of the values $y_i$ are the $y$-coordinate of ...
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An upper and lower bound of $abcd$
Problem. Given $a,\,b,\,c,\,d$ be non-negative real numbers. Denote $$u = \frac{a+b+c+d}{4}, \quad v^2 = \frac{ab+ac+ad+bc+bd+cd}{6}.$$ Prove that
$$12u^2v^2-8u^4-3v^4-8u(u^2-v^2)\sqrt{u^2-v^2} \...
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Find scalar that minimises spectral radius of a matrix
I have a matrix $A$ that is a quadratic function of a real scalar $\beta$ and real constant matrices $B,C,D$:
$$
A = B + \beta C + \beta^2 D
$$
I want to find the value of $\beta$ that minimises the ...
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Prove $\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$
Problem. Let $a,b,c$ are positive real numbers. Prove that
$$\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$$
The inequality is equivalent to $$...
user1656827
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Given a quadratic $f(x) = ax^2 + bx + c$ with $a > 100$, what is the maximum number of integers $x$ satisfying $|f(x)| \leq 50$?
I'm currently stuck on an interesting problem involving quadratic functions. Here's the setup:
Given a quadratic $f(x) = ax^2 + bx + c$ with $a > 100$, what is the maximum number of integers $x$ ...
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Do we need to specify "exception for $x=-5$ and $x=3$" when asking about real roots of equations? [closed]
I want to write a question but I am not sure whether the first version is enough.
Determine the range of $p$ such that $px^2+(2p-1)x -15p+4=0$ has real roots.
Or should I rewrite it as follows:
...
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Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$ [closed]
Consider the equation $(m-1)x^2-(3-m)x-m=0$ with m real numbers $m$ different from $1$, having roots $x_1, x_2$.
Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$.
my ideas
So I was able ...
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If the equation $\ln\left(x^2+5x\right)-\ln(x+a+3)=0$ has exactly one solution for $x$, then possible integral value of $a$ is
If the equation $\ln\left(x^2+5x\right)-\ln(x+a+3)=0$ has exactly one solution for $x$, then find interval of values of $a$
My Approach:
Domain of logarithmic function $\ln(x^2+5x)$ is $x\in (-\infty, ...
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