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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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Is there always a way to simplify $\sqrt{x+y\sqrt{n}}$ in the form of $a+b\sqrt{n}$? [duplicate]

Certain expressions of the form specified in the question can be simplified. $\sqrt{7+4\sqrt{3}}=2+\sqrt{3}$, for example. My attempt: Assumptions: a and b are rational. Assume that $\sqrt{x+y\sqrt{n}...
Alexander Jovanovic's user avatar
1 vote
0 answers
48 views

general quadratic equation with variable coefficients

Consider an equation of the form $a(x)x^2 + b(x)x + c(x) = 0, a(x) \neq 0$. The solutions can be found from the equations $x = \dfrac{-b(x) \pm \sqrt{b^2(x) - 4a(x)c(x)}}{2a(x)}$ which doesn't ...
H-a-y-K's user avatar
  • 729
2 votes
3 answers
97 views

$a/(a + 1)$ and $b/(b + 1)$ are given roots. Find quadratic with roots $a$ and $b$.

$a/(a+ 1)$ and $b/(b + 1)$ are the roots of $x^2 + 7x + 3 = 0$. Find a quadratic with roots $a$ and $b$. In questions like this, I've been tediously simplifying them to equations just having $a + b$ ...
Saransh Arora's user avatar
-4 votes
1 answer
34 views

If $a,b,c \in R$ such that $a < b < c < d$ , show that $(x-a)(x-c) + 2(x-b)(x-d) = 0$ has real and distinct roots . [duplicate]

If $a,b,c \in R$ such that $a < b < c < d$ , show that $(x-a)(x-c) + 2(x-b)(x-d) = 0$ has real and distinct roots . I'm not getting good manipulation idea to show discriminant positive , ...
Ash_Blanc's user avatar
  • 1,089
0 votes
1 answer
83 views

If $n$ is an even and $\alpha, \beta$ are the roots of the equation $x^2 + px + q = 0$ and also of the equation $x^{2n} + p^nx^n + q^n = 0$ ...

If $n$ is an even and $\alpha, \beta$ are the roots of the equation $x^2 + px + q = 0$ and also of the equation $x^{2n} + p^nx^n + q^n = 0$ and $f(x) = \frac{(1+x)^n}{1+x^n}$ where $\alpha^n + \beta^n ...
Ash_Blanc's user avatar
  • 1,089
-2 votes
0 answers
31 views

can anyone tell me what this symbol imply [closed]

It has greater than symbol with less than symbol Image link
Aarav Raj's user avatar
0 votes
4 answers
122 views

Solve $x^2+6x-15120=0$

Solve $x^2+6x-15120=0$ So I could just use the quadratic equation and get the answer. However, I have factored $15120 = 2^4 \cdot 3^3 \cdot 5 \cdot 7$. I know I need two factors with a difference ...
ronald christenkkson's user avatar
2 votes
3 answers
90 views

Finding all real values of $k$ so that the eqn. $(x^2-x+1)^2- 2kx (x^2+x+1) + (x^2+x+1)^2=0...$

Finding all real values of $k$ so that $$f(x)=(x^2-x+1)^2-2kx(x^2+x+1)+(x^2+x+1)^2=0$$ has two or four real roots. We can re-write this equation as $g(y)=y^2+ky+1-k=0$, where $y=\frac{x^2-x+1}{x^2+x+...
Z Ahmed's user avatar
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-2 votes
1 answer
52 views

Let $\alpha, \beta$ be the roots of $x^2-a_1x+1=0$, and consider the sequence of numbers $a_r(r\ge0)$ with $a_0=1$ and $a^2_{r+1}=1+a_r.a_{r+2}$

Let $\alpha, \beta$ are roots of equation $x^2-a_1 x+1=0$ and consider sequence of numbers $a_r,\;r\geq0\;$ with $a_0=1\;$ and $a_{r+1}^2=1+a_r\cdot a_{r+2}.\;$ Then which of the following is/are true?...
aarbee's user avatar
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0 votes
0 answers
52 views

Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns

I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
Kyle Liu's user avatar
0 votes
0 answers
17 views

Question about 2D-Fresnel integration

Thank you for reading my question! I am trying to get the following integration: $$ \begin{align*} \int_0^\infty\int_{-\infty}^{+\infty}e^{j(a(x-x_0)^2+b(y-y_0)^2+c(x-x_0)(y-y_0))}dxdy \end{align*} $$ ...
Xiangyu Cui's user avatar
0 votes
1 answer
30 views

Quadratic where roots and coefficients together form Arithmetic Progression

Background I was reading this post: A.P. terms in a Quadratic equation. And wondered the following: Given a quadratic $ax^2+bx+c=0$ which has roots $x=m,x=n$, is it possible for $a,m,b,n,c$ to be ...
Red Five's user avatar
  • 2,762
1 vote
0 answers
88 views

How to find x if its in the denominator and on the other side of equal sign. [duplicate]

$$\frac{a}{b+xc} = x $$ So I have $x$ in the denominator on the left side and also $x$ on the right. $a$, $b$, and $c$ are known and I need to find out for $x$. How should I go about it? Edit: Thanks ...
Simqer's user avatar
  • 11
0 votes
0 answers
102 views

Ages of Xander, Alice, Bob, and Carol

I need help with quadratics and derivatives for this problem: Problem: Xander, Alice, Bob, and Carol all have unknown ages. We are told that if we multiply the age of Alice by the square of the age of ...
Michael's user avatar
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0 votes
1 answer
21 views

Find the canonical form of each equation and classify the quadratic curve

I am trying to get this equation to the canonical form using Lagrange Method (completing the squares). $$5x^2 + 4xy + 8y^2 - 32x - 56y + 80= 0$$ The first step is to get rid of the $4xy$ term, which ...
Lau-xd's user avatar
  • 1
4 votes
2 answers
472 views

Quadratic with integer roots

This question is from the 1991 Russian Mathematics Olympiad Grade 10 Final round. I have searched as best I can on this site and AoPS for a solution to compare to my own but with no luck. The question ...
Red Five's user avatar
  • 2,762
-2 votes
1 answer
83 views

Why do we need this inequality? [closed]

I have been going through a state maths exam and am not able to answer the following question. I understand that $k>-1/4$. However, in the solutions it is also stated that $e^{-x}>0$. Why does ...
BlingGosling333's user avatar
3 votes
0 answers
64 views

Simplifying a quadratic expression under square root

I am trying to simplify the following expression $$\sqrt{R}:=\sqrt{a^2(u_1-u_2)^2+b^2(u_1+u_2)^2-2 ab (u_1^2+u_2^2-2)}$$ I have been staring at it for a while in the hopes of getting rid of the square ...
RKLS's user avatar
  • 81
1 vote
1 answer
41 views

Quadratic equation with a relation between its coefficients

Given the quadratic equation $ax^2+bx+c=0$, where $a, b, c \in\mathbb{R}$ such that $4(a+b)+7c=0$, $(a\neq0)$ prove that: All of the quadratic's roots are real. Atleast one of the roots is in the $[0,...
fikooo's user avatar
  • 409
1 vote
2 answers
91 views

Quadratic solution from quartic

I'm struggling to follow the derivation in the steps below, which are copied from a textbook (Agarwal, Foundations of Analog and Digital Circuits). I can't follow the derivation, which is solving (14....
Halleff's user avatar
  • 133
0 votes
1 answer
88 views

Is it possible for a quadratic function to not have $y$-intercept?

I know quadratic functions may have up to two $x$-intercepts, but can they have no $y$-intercepts? Edit: I mean quadratic functions which output parabolas (i.e. $f(x)=ax^2+bx+c$ where $a\neq0$). So ...
Vee's user avatar
  • 11
1 vote
1 answer
100 views

Solve the given quadrilateral

A quadrilateral $ABCD$ as $\angle A = 60^\circ, \angle D = 70^\circ, AB = 10$, and $ BC = CD = DA = x $ where $x$ is unknown. Find $x$ and $\angle C$. My attempt: Using the law of cosines on $\...
Quadrics's user avatar
  • 23.9k
4 votes
3 answers
746 views

why is it called quadratic formula? [duplicate]

why is the quadratic formula called the quadratic formula when quad means four and x is only to the power of two and there are only three terms and you can use it more than four times. this is very ...
user avatar
0 votes
0 answers
32 views

Real Life Applications of Two-Variable Quadratic Formulas

Where do two-variable quadratic formulas show up today as real-life combinatorial complexity challenges? Weather? Particle motion? Celestial calculations? Routing? Does anyone have specific examples? ...
Schmyndi's user avatar
4 votes
5 answers
668 views

Discriminant formula - do the coefficients include their signs?

Beginner here. Can I ask someone to explain to me the following? Given the following general form of quadratic equation: $$ ax^2+bx+c=0 $$ and the following formula for discriminant: $$ D=b^2−4ac, $$ ...
DevelBase2's user avatar
1 vote
2 answers
50 views

How to assign > and < signs after finding the roots of a quadratic inequality?

In the reference question, we find the roots of the equation to be $-9$ and $-1$. When we set the quadratic equation to be $> 0$. The values received are $k < -9$ and $k > -1$. Shouldn't this ...
Krishna's user avatar
  • 33
0 votes
1 answer
125 views

If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a function given by $f(x)=\dfrac{(2x-1)(2x^2-4 px+p^3)}{(x+1)(x^2-p^2x+p^2)}$

If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a bijective function defined by $f(x)=\dfrac{(2x-1)(2x^2-4 px+p^3)}{(x+1)(x^2-p^2x+p^2)}$, where $p\geq 0$, then which if the ...
mathophile's user avatar
  • 3,835
7 votes
3 answers
240 views

constraints on the sum and product of roots of quadratic equation assuming less than unity roots

I am solving a math contest problem. Assume we have the quadratic equation $x^2+a_1x+a_2=0$ where $a_1,a_2\in \mathbb{R}$ are real numbers. The roots of this equation can be found as (from equation it ...
K.K.McDonald's user avatar
  • 3,253
3 votes
1 answer
27 views

How does the largest eigenvalue of symmetric matrix associated with a quadratic form change with dimension n.

This is likely not the smartest question but I just wanted to ask if someone could explain why the Lipschitz constant of a quadratic $x^TAx$ where A has entries randomly generated from the Uniform ...
ufghd34's user avatar
  • 71
0 votes
0 answers
26 views

Possibilities for even quadratic

Is the only possibility for an even quadratic $ax^2$ + b where a and b are constants? Also is it necessary for the coefficients to be real?
Sarah's user avatar
  • 9
0 votes
0 answers
124 views

The number of elemements in $\{a\in\mathbb{Z}: |A|\geq4\}\cap\{x\in \mathbb{Z}:x>-10\}$ is equal to

Consider the following set $A=\{x\in \mathbb{Z} : a^3+a^2|x+a|+|a^2x+1|=1, a\in \mathbb{R}\}$. Then the number of elemements in $\{a\in\mathbb{Z}: |A|\geq4\}\cap\{x\in \mathbb{Z}:x>-10\}$ is equal ...
mathophile's user avatar
  • 3,835
3 votes
1 answer
133 views

The quadratic equation $a_1x^2-a_2x+a_3=0$ $a_1, a_2, a_3\in \mathbb N$ has two distinct roots $\in(1,2)$, then what is the least value of $a_1$?

If the quadratic equation $f(x)=a_1x^2-a_2x+a_3=0$  where $a_1, a_2, a_3\in \mathbb N^+$ has two distinct roots belonging to the interval $(1,2)$, then what is the least value of $a_1$? my attempt $D\...
user1318878's user avatar
-1 votes
1 answer
24 views

prove that the point where the gradient of a quadratic function is equal to 0 is always a maximum or minimum point

For context, I am an a level maths student going back through all the content on the course and trying to understand everything that was taught throughout the year with proofs or at the very least, a ...
Morgan's user avatar
  • 31
4 votes
1 answer
438 views

What is the difference between roots and zeroes? [duplicate]

Suppose I have a polynomial of degree 6. It crosses the x-axis at 3 distinct points, and the graph of the polynomial touches the x-axis at one of those 3 points (a repeated root). Question 1: What is ...
Kampann's user avatar
  • 117
0 votes
0 answers
40 views

Non-Euclidean projections

Background Let $y\in\mathbb{R}^2$ be a given point and let $V_1,V_2\in\mathbb{R^2}$ be the vertices of a given segment. Define the projection of $y$ over the segment $s=(V_1,V_2)$ as \begin{equation*} ...
matteogost's user avatar
0 votes
2 answers
46 views

Question regarding sign of a trignometric quadratic functions

This question is regards of the following problem If $\cos^2(x) + (1-c)\cos(x) + 2c - 6 \geq 0$ for every $x \in R$ than the true sets of values of $c$ is, I tried to solve the above problem as ...
koiboi's user avatar
  • 766
0 votes
1 answer
51 views

Solution of the quadratic equation including floor function. [closed]

Let $f(x)=x^2 + (x - \lfloor x\rfloor)*\lfloor x\rfloor - 5$. I wonder that how to solve the equation $f(x)=0$. Is there a theorem that tells whether the equation has a solution? Thanks.
MATIRMAK's user avatar
  • 147
0 votes
0 answers
90 views

Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has at least one root in $[0,3)$ - how to show this?

Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has (A) no root in $[0,3)$ (B) all root in $[0,3)$ (C) exactly one root in $[0,3)$ (D) at least one root in $[0,3)$ In this question I ...
user avatar
-3 votes
1 answer
70 views

Cube root equation [closed]

I'm trying to solve: $$\sqrt[3]{x^2} - \sqrt[3]{x} -6 = 0$$ I’ve tried putting the $-6$ on the other side and cubing both sides but no joy at finding value $x$.
Edgar the Innumerate's user avatar
2 votes
0 answers
43 views

Ordinate and Abscissa of line parallel to tangent of parabola

A parabola is defined by a focus, $F=(p,q)$, and a directrix, $y=l$ (as shown in the diagram). I want to identify the geometric representation of both an ordinate to the diameter, $x=u$, as well as ...
DJ_3629's user avatar
  • 123
-1 votes
1 answer
70 views

Find constants $p$, $q$, and $r$ for $\frac{16x+1}{px+1} > x+4$ where solution set is $x < q$ or $r < x < 3$ [closed]

Find constants $p$, $q$, and $r$ for $$ \frac{16x+1}{px+1} > x+4 $$ where solution set is $x < q$ or $r < x < 3$. Attempted to rearrange for quadratic, but resulted in range of values ...
Hooman's user avatar
  • 51
2 votes
1 answer
84 views

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 ...
ruben alexander's user avatar
0 votes
1 answer
36 views

Quadratic Inequalities solutions

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 ...
Anannyam Loy Barooah's user avatar
8 votes
3 answers
153 views

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-...
Skdmg's user avatar
  • 640
4 votes
3 answers
181 views

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 ...
IONELA BUCIU's user avatar
  • 1,125
0 votes
1 answer
46 views

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 \begin{equation} \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
matteogost's user avatar
4 votes
4 answers
146 views

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 ...
Daksh's user avatar
  • 309
1 vote
0 answers
29 views

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 ...
matteogost's user avatar
0 votes
0 answers
13 views

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 ...
Euler_Salter's user avatar
  • 5,237
-1 votes
1 answer
37 views

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 ...
Red Five's user avatar
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