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 $y=a(x+b)(x+c)$.
5,162
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What will be the value of k for which equation $x^2–4|x|+3=|k−1|$ has four real roots?
The equation is $$x^2–4|x|+3=|k−1|$$
There are several ways to find the solution using either graph or analytically. I want to know is how to do the graphical solution free hand without a calculator. ...
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Quadratic surface where variables are functions
Is it possible for a quadratic surface to have variables and coefficients that are functions? For example, an equation such as
$$A(T)x(T)^2 + B(T)y(T)^2 + C(T)z(T)^2 + D(T)x(T)y(T)\;...$$
where both ...
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1
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Why is $f(x)\equiv f(x+2k)\pmod k$ for$ f(x)=x(x+1)/2 + c.$
I came across this in a programming contest:
f(x)%k = f(x+2k)%k for f(x)=x(x+1)/2 + c
I first thought that this is a property of Harmonic numbers only but after ...
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Is there a general method for solving a parabola in integers? [duplicate]
A parabolic equation such as the following:
$$538445x + 75816 = y^2 $$
with general form:
$$ax + b = y^2 $$
By a brute-force search, one Diophantine solution is $$ x = 2312 $$
Is there a smarter, ...
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2
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How are the domain and range of a quadratic function determined?
How are the domain and range of a quadratic function determined?For example, what are the domain and range of $x^2-4x+10$
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Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
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Randomly sampling of points from the intersection of unit n-sphere and n-dimensional Quadratic form
I want to randomly sample points that have unit norm and lie on a n-dimensional quadratic form
$x^T Ax +b^T x +c =0 $.
$\|x\|=1$
Here, $x$ is a n-vector. $A \in R^{n \times n}$ is a real, symmetric ...
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2
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Positive solutions to $x^2 - 2x - 3 = \sqrt{x+3}$
I found this interesting problem in one of my practice tests
The positive real solution of $x^2-2x-3=\sqrt{x+3}$ has the form of $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, $c$ are primes. Find $a+b+c$.
...
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Lion and wildebeest population modelling question issue
I am a student completing A levels and recently received the following question in an official mock exam for the Edexcel A-level in Mathematics. Having discussed the question extensively with my peers ...
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How can one prove that if the discriminant of a quadratic function is strictly positive, the the quadratic function has distinct roots.
It is well known fact that given $f(x)=ax^2+bx+c, \ a \neq 0$ and $\Delta=b^2-4ac>0$, then $f(x)$ has
two distinct roots.I assume we are in $\mathbb{R}$ at any stage of the problem.
My Attempt
To ...
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How to factor $3x^2-4q^2-3p^2+4qx-8pq$?
I only know the usual techniques for factoring. But I came across this problem:
$$3x^2-4q^2-3p^2+4qx-8pq$$
I couldn't factorise this but thanks to the solution booklet, I found the answer to be:
$$(...
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How do we derive our range of values in plotting graphs of two systems of equation?
This is a topic that has to do with plotting graphs of simultaneously linear and quadratic to find x and y. In most questions or examples, we are usually given range of values as in (A) below while ...
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FFT Bin Estimation - Quadratic Interpolation Equation Differences
Regarding FFT peak Bin estimation, excerpts from Julius O. Smith's SPECTRAL AUDIO SIGNAL PROCESSING and PARSHL detail a derivation for curve fitting a magnitude-maximum FFT bin with its two neighbors, ...
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Finding $x$ in $ax^y - bx^z = c$ where a,y,b,z,c are known [closed]
if values a,y,b,z and c are known, is there any way to find $x$ in equation $ax^y - bx^z = c$ where $y,z \in \mathbb{R}$
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How do you solve two simultaneous quadratic equations using coordinates? [closed]
I know that a quadratic function $ax^2 + bx + c$ passes through the points $(0,0)$, $(1,3)$ and $(5,-11)$
how do I find $a$ and $b$ in the function?
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Geometric interpretation of integral $-\int\frac{1}{\sqrt{a+2bx-hx^{2}}}dx=\frac{1}{\sqrt{h}}\arccos\frac{b-hx}{\sqrt{b^{2}+ah}}$
The following formula is given as "the familiar arc-cosine form" by Joos, in his Theoretical Physics. The German language original has $e$ in place of $h$.
$$-\int\frac{1}{\sqrt{a+2bx-hx^{2}...
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$p$ and $q$ are solutions of the equation $x^2 +p*x+ q=0$. Find values $p$ and $q$.
Given that $p,q$ are roots of the equation $x^2+p*x+q=0$. Find values $p$ and $q$.
One method of finding a solution is using Viète’s Theorem.
So, $p+q=-p$ and $p*q=q$ and there are two solutions $p=0, ...
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Given $a + b + c = 20$, find $\max(ab + ac + bc)$
Given that $a + b + c = 20$, what's the maximum possible value for $ab +ac + bc$?
$$\left(a, b, c \in \mathbb{N}\right)$$
I tried following this post, Evaluating max(ab+bc+ac), which lead to:
$ab + ...
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2
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Range of $f(x)=x \sqrt{1-x^2}$
I have to find the range of $f(x)=x\sqrt{1-x^2}$ on the interval $[-1,1]$. I have done so by setting $x=\sin\theta$ and thus finding it to be $[-0.5,0.5]$.
Let $x=\sinθ$. Then, for $x\in[-1,1]$ we get ...
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Determine a value, where quadratic equation will generate the perfect square
In order to find RSA factors ($pq = N$), we have to solve a quadratic equation $x^2+Bx+C=y^2$, where:
$x_1 < x_2$,
$x_1, x_2$ and $y_1, y_2$ are (positive) integer numbers,
$x_1$ is the smallest ...
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0
answers
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Graphing Quadratic Equation Number with x
Equation
(7 + 4x)(2x - 4)
Standard Form (Solved)
8x^2 - 2x - 28
I am trying to graph this equation as practice (For School Algebra) And I get this equation in my random generator, So then I go to ...
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3
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How do I solve the equation $(x - 1)(x - 2)(x - 3) = (x - 2)(x - 3)(x - 4)$?
Problem: $(x - 1)(x - 2)(x - 3) = (x - 2)(x - 3)(x - 4)$
Heres my question with this problem: why do I end up with a wrong answer when I divide both sides by $(x-2)(x-3)$ to cancel out the $(x-2)(x-3)$...
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Calculate $\frac{2}{\alpha^2}-\frac{1}{(\alpha +1)^2}$ if $\alpha$ be a root of $x^2+(1-\sqrt3)x+1-\sqrt3=0$
If $\alpha$ is a root of $x^2+(1-\sqrt3)x+1-\sqrt3=0$ calculate
$$\frac{2}{\alpha^2}-\frac{1}{(\alpha +1)^2}$$
What I have done:
$$\begin{aligned}
\alpha^2+\alpha+1&=\sqrt3(\alpha+1)\\
&\...
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How can I integrate $\int\sqrt{\frac{x^2+bx+c}{x^2+ex+f}}dx?$
How can I integrate
$$\int\sqrt{\dfrac{x^2+bx+c}{x^2+ex+f}}\,dx?$$ I was thinking a substitution $$t=\frac{x^2+bx+c}{x^2+ex+f},$$ which inverts as follows:
$$(x^2+ex+f)t=x^2+bx+c$$
$$(t-1)x^2+(et-b)x+...
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How to maximize profit using a non-linear demand equation?
I'm working on a project involving quadratic equations and I'm supposed to use them to determine the optimal selling price and quantity for a product.
I've tried using a linear demand equation and ...
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3
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Confusion while factoring quadratics.
I recently factored the quadratic $3x^2+4x-4$, the results where $x = -2$; $x = 2/3$. My question is why is it incorrect to write the factors of the quadratic as $(x+2)(x-2/3)$? The correct way to ...
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Proving this by quadratic function
Given are $x$, $y$ and $z\ge0$, if $x^2+y^2+z^2+2xyz=1$, show that
\[x+y+z\le\dfrac32.\]
I want to solve this by quadratic function.
Let $x+y+z=k$, so $z=k-x-y$. Plug in and expand to have
\[-2 x {{y}...
user1034536
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How to find quadratic equation when the complex roots are given?
For example, $x= -4+0.4i$, $x= -4 - 0.4i$
How do I find remove the complex $i$ and find the quadratic formula ?
Someone told me that the formula is $x^2$ - (sum of roots)$x$ + (product of roots) = $0$...
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Early 1900 algorithms for factoring quadratic trinomials where a >1
I've been reading two algebra textbooks from the early 1900's. Each presents an algorithm for factoring quadratic trinomials when the coefficient of the $x^2$ term does not equal one. The technique ...
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How to find possible equations given 2 points and the $y$-value of a vertex
If I have 2 points (let’s say $(0,8)$ and $(6,0)$) and a line on which a vertex can be ($f(x)=16$), how can I find the possible quadratic equations that would intersect both the points and have the ...
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1
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Proving no integer solution exists that makes a polynomial a perfect square
The context for this is the following coding problem on Hackerrank. I'm trying to understand why one of their sample inputs (Sample Input 4) has no solution.
After a bit of math, it comes down to ...
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Approach to solving a unitary method problem
I saw this question in a textbook:
Two taps fill a container in $\frac{75}{8}$ hours together. If one tap takes $10$ hours more than the other, how long does each tap take to fill up the container?
...
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Find the integer integers k for which there exists an integer x [closed]
Find the integer integers k for which there exists an integer x
$\sqrt{39-6\sqrt{12}}+\sqrt{kx(kx+\sqrt{12}+3)}=2k$
So far I haven't advanced much. Removing brackets didn't do anything for me and was ...
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Non-Monogenic Quadratic Extensions
Fix a quadratic extension of global fields $L/K$; i.e., $[L:K]=2$. Must there exist some $\alpha\in\mathcal O_L$ such that $\mathcal O_L=\mathcal O_K[\alpha]$?
Of course, the answer is positive for $K=...
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What is the isotropic cone of det in $M_2(K)$?
Consider the application $\text{det}: M_2(K) \to K$
What is the isotropic cone of det in $M_2(K)$?
Consider the vector subspace $D$ of the diagonal matrices of
$M_2(K)$. Is it true that $M_2(K)=D \...
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For what integers $n$, do we have $30n+11=6x^2+5y^2$ for some integers $x,y$?
I am trying to find all integers $m$ such that $m$ is relatively prime to 30, and $m=6x^2+5y^2$ for some integers $x,y$. Note that we must have: $y$ is odd, $(y,3)=1=(x,5)$. Using these conditions, I ...
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When do two quadratic equations have exactly one common solution?
I have stumbled upon the following two exercises:
For $a,b \in \mathbb{Z}$, consider $A=\{x\in\mathbb{R} | x^2+2ax+b=0\}$ and $B=\{x\in\mathbb{R} | x^2+2bx+a = 0\}$. Let $k$ be the number of elements ...
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A proof involving a quadratic
Assume we have the equation:
$$c = (a-n)(b-n)$$
where a, b and c are constants.
Solving for <...
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Equation for rotated elliptic Parbolloid
I am trying to do a manual Chisquare fit on a 2D data, my problem is that my data shows a rotated elliptical parabolloid. For that purpose I use the equation:
$$f(a,par)=offset + ((a - a_{0}) Cos[\...
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Is it possible that $mx^2-x+1=0$ has two negative roots?
If the expression $(mx-1+\frac1x)$ is non-negative for all positive real $x$, then find the minimum value of $m$.
Given, $\frac{mx^2-x+1}x\ge0$ for all positive real $x$.
$\implies mx^2-x+1\ge0$ for ...
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2
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If both side of the equation have variable and we take the square root of both sides, do we add the plus and minus sign?
For example:
$a^2 = b^2 + c^2$ with $a, b$ and $c$ are real numbers.
$a = \sqrt{b^2 + c^2}$
$a = ±\sqrt{b^2 + c^2}$
Should the answer be 1 or 2?
I know this sounds obvious, but I ask because in ...
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Parabola where the $x$ intersect is always $0$ and $1$
I am working on creating a parabola where the $x$ intersect is always $0$ and $1$ for basic needs the one below works great!
$$
y=-d\cdot a^{2}\left(x\ -.5\right)^{a}\ +d
$$
but I am attempting to ...
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0
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Domain of Real Roots of a Quadratic
The following MCQ(single correct option) is from JEE Main 2022
Let $S$ be the set of all integral values of $\alpha$ for which the sum of squares of two real roots of the quadratic equation $3x^2+(\...
- 949
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2
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Quadratic equation problem
Nino sells on average three cell phones more than Will. At the end of the day, they both receive $35.00$ together for the day's sales. The store manager finds that: If Will sold at price W the number ...
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4
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Contest Math Question on Logarithms
I am trying to solve a question from the AoPS Vol. 2 book.
The question is as follows:
Suppose the $p$ and $q$ are positive numbers for which:
$$\log_9 p = \log_{12}q = \log_{16}(p+q)$$
What is the ...
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2
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I am having trouble finding the nth term of a sequence. I understand how to find the second difference but the rule with A confuses me.
There are $6$ terms listed from term $1$ to term $6$ as:
$1,3,6,10,15,21$
from this came the first difference:
$2,3,4,5,6$
and the second difference is:
$1,1,1,1$
Does this change anything with the ...
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vote
0
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Inequality for solutions a solution of system of quadratic equations.
Let $v_{i,j}\in [0,1], i,j = 0,\ldots,n$ solve the following system of equations for $r>0$:
\begin{align}
0 &= \tfrac 1 2 e_{i,j}^2 + e_{j,i} v_{i,j+1} - (r+e_{j,i}) v_{i,j}, \tag{*} \\
e_{...
- 1,351
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1
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Quadratic Diophantine equation [duplicate]
how one can approach solving diophantine equation $a^{2}$+ $b^{2}$ - $1$ = $n {\cdot}a{\cdot}b$ in positive integers $a$, $b$, $n$ ( where also $a$ and $b$ are relatively prime)? I have tried so far ...
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2
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2
answers
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Finding (approximate) intersection of logarithm and quadratic functions
I am trying to find a closed form solution to an equation of the form $$x^2+ax+b=c\log x$$
I found some resources that express solutions in terms of the Lambert W function if the LHS is linear but I ...
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2
answers
49
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Logs going through a quadratic isn't a valid solution path?
Consider $$\log_4(x+5)+\log_4(x+11)=2$$
Raising $4$ to each term across the board and treating the addition as a multiplication you get a simple quadratic, whose roots are $x=\{-3,-13\}$. Easy Peasy. ...
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