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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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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
32 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
-6 votes
0 answers
42 views

Probability and Sum of All Solutions [closed]

Given x=5.5386388265268286537811327028332 Then 2x^2 - 3x + 1 = 0 or 2*5.5386388265268286537811327028332^2 - 3 * 5.5386388265268286537811327028332x + 1 = 0 The question Im asking, does x hold true for ...
AllTBCreative's user avatar
0 votes
0 answers
22 views

Finding solution fby changing ellipses [closed]

Lets say we have two, $2$ dimensional positive integer vectors $X_0$ and $X_1$. We know that their sum is $[P,P]= X0+X1$. Lets also say assume that We get $Z=X1-X0$ from the intersection of $2$ ...
Aravind Muraleedharan's user avatar
9 votes
3 answers
126 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
  • 862
4 votes
3 answers
174 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
  • 1,387
0 votes
1 answer
42 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
126 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
  • 97
-2 votes
2 answers
69 views

compose a quadratic equation with integer coefficients [closed]

compose a quadratic equation with integer coefficients having roots $\frac{2-x_1}{x_2}$ and $\frac{2-x_2}{x_1}$, where $x_1$ and $x_2$ are the roots of the equation $3x^2+2x-9=0$
Гоша Рудковский's user avatar
1 vote
0 answers
28 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
11 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
  • 5,173
-4 votes
1 answer
29 views

How would you go about graphing a parabola on a graph and determining what the $x$ is [closed]

Example $-x^2-2=0$. How would you determine this $x$ and how to graph it, is there a formula you must follow? *edit this is confusing me very badly all I've seen was the question like that and it ...
liam's user avatar
  • 9
0 votes
1 answer
31 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
  • 2,064
0 votes
2 answers
45 views

Question regarding max and min of function when derivative cannot be 0

This question is in regards to finding the range of the following function $$x^3 - x^2 + x + 1$$ I decided to find the maxima and minima of this function by computing its derivative, the derivative of ...
koiboi's user avatar
  • 734
0 votes
1 answer
39 views

Is this Result on Continuity of Composite functions true?

Consider, A real valued function $f(x)$ which is discontinuous $\hspace{2mm}\forall \hspace{2mm} x \in [\alpha,\beta]$ and another real valued function $g(x)$ which is discontinuous $\hspace{2mm}\...
Jesko's user avatar
  • 29
0 votes
1 answer
41 views

What are the x intercepts (wherever defined) for $ \vert x^2 -8x +15 \vert ^ {(x-1)(x-2)(x-3)(x-5) \over (x-2)} = 1 $

The original question asked was the following : (Source : https://drive.google.com/file/d/1U7khRKh12GlaSY73210oQd2170JiXEGY/view) Rita took some of her friends for picnic. Her friends are at x−...
Devanshu Kashyap's user avatar
4 votes
4 answers
96 views

the quadratic equation $x^2+(a+b) x+c=0$ has no real roots

If $a, b, c \in \mathbb{R}$ and the quadratic equation $x^2+(a+b) x+c=0$ has no real roots, then (A) $c(a+b+c)>0$ (B) $c+c(a+b+c)>0$ (C) $c-c(a+b+c)>0$ (D) $c(a+b-c)>0$ By ...
math_learner's user avatar
4 votes
1 answer
460 views

Why is it that in some problems that involve systems of equations, solving for one unknown value, also yields the other in the second solution?

Like take the following scenario as an example: Two spherical objects have a combined mass of 200 kg. The gravitational attraction between them is $7.34 × 10^{−6}$ N when their centers are 15.0 cm ...
3h6_1's user avatar
  • 43
2 votes
0 answers
81 views

Graphical obviousification of the quadratic formula [duplicate]

So there is a million (essentially equivalent) ways to make it obvious algebraically that the axis of symmetry of the parabola $y=x^2+bx+c$, must be at $x=-\frac{b}{2}$, while its real roots, if they ...
Damian Reding's user avatar
4 votes
3 answers
220 views

Help with the indefinite integral $\int \frac{dx}{2x^4 + 3x^2 + 5}$

I start by rewriting the denominator, $2x^4+3x^2+5$, as a squared term plus a constant. To do this, we notice that the first two terms already have a common factor of $2x^2$. We can complete the ...
Meharaj hossain Arman's user avatar
0 votes
2 answers
42 views

Question regarding formula for range of quadratic function

While reading through my textbook i saw 2 formulas for the range of quadratic functions as follows $$\text{When } a > 0 \text{ range is } \left[\frac{-D}{4a}, \infty\right)$$ $$\text{When } a < ...
koiboi's user avatar
  • 734
1 vote
1 answer
55 views

Navigating Consecutive Integer Quadratic Roots:

Prove that if $a$,$b$ and $c$ are consecutive positive integers then the roots of the quadratic equation of the form $ax^2 + bx + c = 0$ has Complex roots My attempt: In my attempt to grapple with ...
StudyME's user avatar
  • 27
0 votes
1 answer
63 views

If roots of $a(x^2+m^2)+amx+cm^2x^2=0$ are $u,v$

If roots of $$a(x^2+m^2)+amx+cm^2x^2=0.......(1)$$ are $u,v$, one can easily prove that $$a(u^2+v^2)+auv+cu^2v^2=0.........(2)$$ Interestingly, if (1) is $f(x,m)=0$, then (2) is $f(u,v)=0.$ The ...
Z Ahmed's user avatar
  • 43.2k
2 votes
2 answers
212 views

Proof related to factorization of a quadratic equation

Can we prove or disprove this given statement: "If a, b, and c are integers with a≠0, and the roots of the equation $$ax^2 +bx+c=0$$ are rational, then the equation can be factored as $$(ax+m)(x+...
Briston's user avatar
  • 156
0 votes
0 answers
13 views

How do you find the average rate of change over an interval that is a compound inequality?

Let $f(x)=0.7x^2$ and $g(x)=2.8x^2$. How do I find the average rate of change over $f(x)$ and $g(x)$ over the interval $[4,8]$?
blegh's user avatar
  • 1
0 votes
0 answers
47 views

General formula for factoring $ax^2+bx+c$ [duplicate]

I watched a YouTube video on factoring (namely, 100 trinomial factoring (Dedicated to Mr. Hill) by blackpenredpen) and was curious about the math behind the method he first shows around 12:15. Is ...
voltedz's user avatar
0 votes
1 answer
44 views

What is the sum of all the roots of the equation $x|x|-5|x+2|+6=0$?

Question: What is the sum of all the roots of the equation $x|x|-5|x+2|+6=0$? My Method of Solving: 3 cases arise here. Case 1: Considering that $x\geq0$, the equation becomes $x^2-5(x+2)+6=x^2-5x-4=...
Harikrishnan M's user avatar
0 votes
1 answer
28 views

Quadratics - Sum of areas is minimum...

I am trying to help my daughter with this question: ...
Rosie Smith's user avatar
0 votes
0 answers
40 views

Maximising a quadratic expression in 3 variables

If $x^2 + y^2 + z^2 = 1$ Maximise $(cy-bz)^2 + (az-cx)^2 + (bx-ay)^2$ This can be written as the square of the magnitude of the determinant: $$ \begin{vmatrix} \widehat{i} & \widehat{j} & \...
randoLorries's user avatar
0 votes
2 answers
128 views

Find the maximum value of $m^2+n^2$ if $(m^2-mn-n^2)^2=1$

Given that the integers $m$ and $n$ in the set $A=\left\{1,2,3,....,2024\right\}$ satisfy $(m^2-mn-n^2)^2=1$. Find the maximum possible value of $m^2+n^2$. My effort: We have $m^2-mn-n^2=\pm 1$ Case $...
LifeIsMath's user avatar
0 votes
0 answers
32 views

How to proceed in order to prove the equality conditions mentioned?

My proof is given below : $\displaystyle a\cos^{2} x\ +\ b\cos x\ +\ c\ =\ 0$ is the given equation. We know, $\displaystyle \cos 2x\ =\ 2\cos^{2\ } x\ -\ 1$ and from this we can express $\...
Aritro Shome's user avatar
0 votes
1 answer
58 views

A question about derivative and polynomial

Given a polynomial $p(x) = x^3 + kx - 2$, where $k \in \mathbb{R}$ is a constant and $p(x)$ has a double root at $x = \alpha$. Prove that $\alpha=-1$ and $k=-3$. I plugged in $\alpha$ for $p(x)$ and $...
LÜHECCHEgon's user avatar
0 votes
0 answers
12 views

How to get the solution of (\alpha Z-\ Omega) from the following two equations?

I have the following two equations $$-\frac{\gamma+2P}{1+2Z}[Z^2+(\alpha Z-\Omega)^2]+2(P-\gamma z)[Z+\alpha(\alpha Z-\Omega)]=0$$ (1) $$2Z[-2\gamma_1\frac{\gamma+2P}{1+2Z}Z+Z^2+(\alpha Z-\Omega)^2]-...
Udichi's user avatar
  • 123
6 votes
1 answer
137 views

Are there general solutions to quadratic, 2D, continuous, time-invariant dynamical systems?

I am a bit new to dynamical systems and don't know my way around terminology, so have had a hard time answering this for myself. I know the basics of theory for 2D linear, time-invariant systems, i.e.,...
dang's user avatar
  • 105
0 votes
0 answers
23 views

Graphing machine for a quadratic equation representing braking speed of a vehicle over time

I am solving for a particular situation. A two vehicles approach an intersection and are on a collision course. To avoid the collision, one vehicle needs to apply the brakes such that it slows to ...
Nathan Legoman's user avatar
3 votes
0 answers
83 views

Ruling out finite time explosions to infinity of Riccati differential equations

Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions. Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive ...
cfp's user avatar
  • 685
0 votes
2 answers
73 views

If $A$ is a symmetric positive definite matrix, show that $f(x) = x^TAx$ is convex.

Let: \begin{gather*} f: \mathbb{R}^n \to \mathbb{R}, \quad A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, x \in \mathbb{R}^n, c \in \mathbb{R} \\ f(x) = x^T A x + b^T x + c \\ \end{gather*} If ...
clay's user avatar
  • 2,713
0 votes
1 answer
67 views

I can't seem to understand the proof of the Cauchy-Schwarz inequality

The proof said to use the quadratic polynomial: $$P(t) = \sum_{i=1}^n(a_it - b_i)^2$$ by which we notice that $P(t) \ge 0$ , but then this is the part which I didn't understand where we conclude that ...
Aymane's user avatar
  • 3
1 vote
0 answers
51 views

Prove or give counter example of quadratic inequality

I have two finite probability mass functions (pmfs) $P(x)$ and $Q(x)$ on the same support $(0,1,\ldots,n)$. Let $(p_0,p_1,\ldots,p_n)$ and $(q_0,q_1,\ldots,q_n)$ be the probability vectors from the ...
user2961927's user avatar
1 vote
0 answers
36 views

How do you rearrange an equation with the product of floor functions?

Edit: I spent some time trying to generalise this formula for any build plate $l$ and $w$ and have realised a very small error in one of the quadratic equations below which I will correct tomorrow ...
Jamie's user avatar
  • 21
-1 votes
3 answers
93 views

Factoring $x^2 + 4x -3$ step by step.

I have the following equation that I need to factor: $x^2 + 4x -3$. I cannot use the factoring by grouping method as there are no integers that add up to $4$ and give $-3$ when multiplied. What method ...
Paul's user avatar
  • 1
1 vote
1 answer
70 views

How to find factors of a number that add to a certain sum?

For example, when finding the roots of this quadratic equation: $x^{2}+8x-9=0$ I would write $x^2-x+9x-9=0$ Then write that expression as the product of two linear expressions: $(x+9)(x-1)=0$ Then I ...
Will Fitchet's user avatar
1 vote
4 answers
159 views

Why does $\sqrt{3}x^2-4\sqrt{3}=0$ not follow discriminant rules?

From what I understand: $D > 0$ and a perfect square $\Longrightarrow$ Real and Rational Roots $D > 0$ but not a perfect square $\Longrightarrow$ Real and Irrational Roots $D = 0$ $\...
Ox Ph's user avatar
  • 31
2 votes
1 answer
42 views

Intersection of parabola and line

Suppose that a parabola and line are given by $$y = 2x-k, y = x^2-(k+2)x+2k$$ and if one of the points at which they intersect is on $x$-axis, how can we find the ordinate of the other intersection ...
user avatar
1 vote
0 answers
79 views

Multiply two lines together

Excuse me , I had a strange question :). In algebraic geometry, a quadratic expression can be written as the product of two different linear equations, for example: y = x² + 3x - 4 can be factored as ...
Mostafa Zeinodini's user avatar
0 votes
2 answers
48 views

Proving a quadratic function is bijective in given range

Prove that $f: [0,\infty[ \to [-5,\infty[$ defined as $f(x) = 4x^2+4x-5$ is bijective. I can prove it graphically but not algebraically.
AshKetchum4441's user avatar
0 votes
0 answers
46 views

Upper bound of diagonal matrix multiplied with Hermitian matrix

Suppose that ${\bf R} \in \mathbb{C}^{n \times n}$ is a Hermitian matrix, and ${\bf D}$ is a diagonal matrix with main diagonal being ${\bf d} \in \mathbb{C}^{n \times 1}$. I am looking for the ...
H. H.'s user avatar
  • 41
2 votes
1 answer
40 views

Can there be more solutions to following equation?

Consider the following equation for $t > 1$: $$(t + \sqrt{t^2 - 1})^{x^2 - 2x} + (t - \sqrt{t^2 - 1})^{x^2 - 2x} = 2t$$ If we let $u = t + \sqrt{t^2 - 1}$ then $\frac{1}{u} = t - \sqrt{t^2 - 1}$ ...
Shiv's user avatar
  • 155
-1 votes
1 answer
88 views

How can I find the other root of a quadratic equation with 1 root and a unknown b value? [closed]

The information I was given was, "If $6x^2 + kx -42 = 0$ has one root as $\frac{-7}{6}$, then what is the other root?
Aaryan Velluri's user avatar
1 vote
0 answers
70 views

Quadratric modular equation [duplicate]

I need to find all numbers \begin{equation} 6x^2 + 2x \equiv 20 \pmod{513} \end{equation} and I'm having a trouble since 513 is not prime number I tried looking for roots but i end up with \begin{...
Karol Bargieł's user avatar

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