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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 $y=a(x+b)(x+c)$.

0
votes
2answers
33 views

Relationship between Discriminant and coefficients of a Quadratic function

To answer question (b), the problem requires me to use the discriminant. However, I still do not understand how would I realize that I would have to make use of it. Can someone please explain the ...
0
votes
1answer
27 views

Finding minimum value of relation with quadratic coefficient

Let $f(x)= ax^2 +bx+c$, such that $f(x)\geq0$. Find minimum value of $\frac{f(1)}{f(0)-f(-1)}.$ My attempt: denominator should be maximum so $f(-1)=0$ as $f(x)$ is always greater than or equal to ...
1
vote
5answers
52 views

What causes the extraneous root when intersecting parabola $y^2=4ax$ with circle $x^2+y^2=9a^2/4$?

If I solve the parabola: $y^2=4ax$ and the circle: $x^2+y^2=\frac{9a^2}{4}$ I get a quadratic in $x$; ie: $$4x^2+16ax-9a^2=0$$ which has roots $x=\frac{a}{2},-\frac{9a}{2}$. But if we see the graphs ...
0
votes
1answer
53 views

Number of different quadratic functions mod 12.

How many different quadratic functions are there of form: $x \mapsto ax^2+bx+c \pmod{12}$ All I could come up with is an upper bound of $11*12*12$. This is a puzzle from a YouTube video by ...
-4
votes
2answers
56 views

If $x^2 +135= x^2 −12$ , then what is the value of $x$? By considering ring. [closed]

How to find the value of $x$ in this types of equations, as I know no real solutions exist, but how to find imaginary solutions
2
votes
5answers
139 views

Solving $x^2-2x-3<0$

If i have to solve $x^2-2x-3<0$ I would do $$x+1 < 0, \quad x-3<0$$ and end up getting $x<-1$ and $x<3$. Why is it wrong to use the same inequality sign? Shouldn’t both of the ...
0
votes
1answer
43 views

Solving in terms of z , three variable two equation system

Solve in terms of $z$ $$ \begin{cases} 4z&= x + 2y \\ 3z^2&=\frac{1}{2}x^2 + y^2 \\ \end{cases} $$ Solution: $x = 2z/3$ and $y = 5z/3$. I don't understand how they got to the solution with ...
0
votes
5answers
60 views

Find the remainder when $p(x)$ is divided by $x^2-a^2$ if $p(x)$ leaves remainders $a, -a$ when divided by $x+a, x-a$

Let $a\neq0$ and $p(x)$ be a polynomial of degree greater than $2$. If $p(x)$ leaves remainders $a$ and $-a$ when divided respectively by $x+a$ and $x-a$. Find the remainder when $p(x)$ is divided by $...
6
votes
4answers
788 views

How to quickly solve partial fractions equation?

Often I am dealing with an integral of let's say: $$\int\frac{dt}{(t-2)(t+3)}$$ or $$\int \frac{dt}{t(t-4)}$$ or to make this a more general case in which I am interested the most: $$\int \frac{...
0
votes
3answers
46 views

Points on ellipse at maximal distance from center [closed]

Suppose we have an equation of an ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = c. $$ My question is where are the points situated on the ellipse at maximal distance from its center? I imagine this ...
0
votes
4answers
44 views

Show that if $ \sigma $ is a solution to $ x^2 + x + 1 = 0 $ then the following equality occurs.

Show that if $ \sigma $ is a solution to the equation $ x^2 + x + 1 = 0 $ then the following equality occurs: $$ (a +b\sigma + c\sigma^2)(a + b\sigma^2 + c\sigma) \geq 0 $$ I looked at the solution ...
4
votes
2answers
62 views

Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots.

Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots. Let $a,b $ be the solutions of second equation, then by Vieta we have $a+...
3
votes
4answers
82 views

Finding the solution set for a quadratic inequality $x^2-2<\frac{7}{2}x$

If $x^2-2<\frac{7}{2}x$ then what is the solution set for $x$? I have most of the problem done, I just don't know how to lay out my answer. The answer is supposed to be $$\boxed{-\frac12<x<4}...
0
votes
2answers
292 views

Solve $(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}=10$

Find the real values of $x$ which satisfy the equation $(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}=10$ My Attempt $$ e^{10}=e^{(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}}=e^{(5+2\sqrt{6})^{x^2-3}}\...
0
votes
1answer
32 views

Integer Solutions for a Multivariable Quadratic

I'm solving a math puzzle and arrived at a quadratic:$$ \frac{6000n}{x(x-n)}=c $$ I only just graduated from high school and have very limited knowledge. I'm wondering if it's possible to find all ...
0
votes
1answer
33 views

Tool/algorithmic library for determining equation of quadric surface through 9 given points

Is there any software that can calculate(and eventually plot) the quadric surface generated by 9 given points in 3D space? I know I can calculate that by defining 9 equations in Mathematica and ...
1
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3answers
52 views

Numbers of roots in Quadratic Equations

If $a,b,c,d$ are real numbers, then show that the equation $$(x^2 +ax -3b)(x^2-cx+b)(x^2 -dx +2b)=0$$ has at least two real roots.
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3answers
28 views

Prove atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots if $b_1b_2=2(c_1+c_2)$

If $b_1b_2=2(c_1+c_2)$, then prove that atleast one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has real roots. $$ \Delta_1.\Delta_2=(b_1^2-4c_1)(b_2^2-4c_2)=b_1^2b_2^2-4b_2^2c_1-4b_1^2c_2+...
4
votes
3answers
177 views

$1$ as a common root of a quadratic equation

$$ax^2 + bx + c = 0\quad\text{and}\quad bx^2 + cx + a = 0$$ have a common root. In my book, it says that 1 is a common root for those equation? Is this correct. When we plug 1 in both the ...
0
votes
4answers
59 views

$x^2-2mx+m^2-1=0$ Find the range of m when one root lies in (-2, 4)

Let: $x^2-2mx+m^2-1=0$. For all $m$ there exist real roots for the above equation. If one root lies in between $(-2,4)$ Find the value range for $m$ My Work Since coefficient of $x^2$ is 1 the graph ...
3
votes
2answers
39 views

For what integral values of $a$ does $x^2-x(1-a) - (a+2)=0$ have integral roots?

For what integral values of $a$ does the equation $$x^2-x(1-a) - (a+2)=0$$ have integral roots? I have tried making the discriminant to be a perfect square; but when it becomes perfect square, $...
1
vote
1answer
47 views

Solving differential equation by factorization

I'm trying to solve the following differential equation: $ (x^{2} -1) y'' - (4x^{2} -3x-5) y' + (4x^{2} -6x -5) y = e^{2x}$ By the method of factorizing operators, so I rewrite the differential ...
1
vote
2answers
29 views

Finding solutions to a complex polynomial

I have the quadratic equation $x^2-5x+7-i=0$, I am not sure how to solve? I was going to use the quadratic formula but I wasn't sure as the quadratic equation seems quite easy and working out $\sqrt{-...
3
votes
4answers
54 views

Finding values for K such that the roots of the quadratic are strictly imaginary: $x^2+\left(K^2+3K-7\right)x+K$

$x^2+\left(K^2+3K-7\right)x+K$ I'd like to know the general approach needed to find out how to find solutions for K when I want the roots of this equation to have a specific property, such as ...
1
vote
2answers
54 views

Find the values of $a,b \in \Bbb R$ (if exists) such that $-5 \le \frac{x^2+ax+b}{x^2+2x+3} \le 4$ for all $x \in \Bbb R$

Find the values of $a,b \in \Bbb R$ (if exists) such that $$-5 \le \frac{x^2+ax+b}{x^2+2x+3} \le 4$$ for all $x \in \Bbb R$ My try: I noticed that $x^2+2x+3 > 0$, so i can divide the inequality ...
2
votes
1answer
41 views

Find (a,b,c) if $ax^2-bx+c=0$ have roots lying in $(0,1)$ where $a,b,c\in\mathbb{Z_+}$ [duplicate]

Let the equation $ax^2-bx+c=0$ have distinct real roots both lying in the open interval $(0,1)$ where $a,b,c$ are given to be positive integers. Then the value of the ordered triplet $(a,b,c)$ can be ...
3
votes
1answer
44 views

Derivative of a multivariate quadratic function

Let's define function $f : \mathbb{R}^n \to \mathbb{R}$ as $$ f(x) = {1\over2}x'Ax + b'x $$ where matrix $A \in \mathbb{R}^{n\times n}$ and vector $b\in \mathbb{R}^n$ are given. Function $f$ is ...
0
votes
3answers
39 views

In solving Real World Quadratic Equations, why does 𝑐 change value, and when do i have to change value? [closed]

When solving for the equation I entered 32, not -32. When I watched the video paired with this course it did not explain in what cases to do this. All help is appreciated :D
4
votes
1answer
64 views

If a,b,c are positive rational numbers such that a>b>c then tell which of the following statement are correct following quadratic equation

I am solving following question based on quadratic equation If $a,b,c$ are positive rational numbers such that $a>b>c$ and the quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root ...
1
vote
4answers
131 views

minimum of $a^2+4b^2+c^2$ given $2a+b+3c=20$

If $a,b,c\in\mathbb{R}$ and $2a+b+3c=20.$ Then minimum value of $a^2+4b^2+c^2$ is what i try Cauchy schwarz inequality $$(a^2+(2b)^2+c^2)(2^2+\frac{1}{2^2}+3^2)\geq (2a+b+3c)^2$$ How do i solve ...
2
votes
2answers
67 views

Is value of $\alpha$ defined?

Consider three quadratic functions: $$P_1(x)=ax^2-bx-c$$ $$P_2(x)=bx^2-cx-a$$ $$P_3(x)=cx^2-ax-b$$ Where $a,b,c \in \mathbb{R}\backslash \left\{0\right\}$ If there exists real number $\alpha$ such ...
1
vote
1answer
49 views

What is the range of $(f+g)(x)$ where $f(x)=x^2+4x-3$ and $g(x)=3x^2-8x+9$?

What is the range of $(f+g)(x)$? I plugged in the domain values to get the range for each of the equations and then I would have summed them up. But I'm getting erroneous results. Because each ...
0
votes
0answers
26 views

Finding how many intersections two parabolas have within a certain domain

I need to find out if $y=x^2-\sqrt{200}x+50$ and $y=a(x-\sqrt8)^2$ how for what values of $a$ they will have only one point of intersection within the domain $[\sqrt8,\infty)$?
1
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1answer
98 views

On monotonic quadratic least squares

Quadratic least squares can be used to fit a quadratic curve to $3$ or more points, such that the resulting curve is the quadratic curve that has the least squared distance of the data points to the ...
0
votes
4answers
53 views

how to memorize the sum and product of roots for an $n^{th}$ degree equation

For my exams I need to know the following equations by heart: for a polynomial equation: $a_nx+a_{n-1}x^{n-1}+...+a_1x+a_0,$ the sum and product of the roots are given by $$\textrm{Sum}=-\frac{a_{n-...
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votes
2answers
79 views

Analytical solution to polynomial system

I have a polynomial system with three equations and three unknowns i wish to solve analytically. I can obtain a numerical solution easily but for my project i need to find a analytical solution. The ...
0
votes
1answer
52 views

How to factor $-x^2 + x -10$?

This question has been killing me for hours. None of the factors of $10$ (because $(-10)(-1) = 10$) add up to $1$. So how do you do this question?
0
votes
0answers
23 views

Quadratic equation, absolute value of roots strictly superior to 1 conditions

Let's consider the equation: \begin{align} 1 - \phi_1 z - \phi_2 z^2 = 0 \end{align} We want to find the conditions on $\phi_1, \phi_2$ for the roots to have an absolute value strictly superior to 1. ...
0
votes
2answers
29 views

Solving quadratic equation for inverse variable

I'm reading through some lecture notes and they show a quadratic equation, which I will just write in the usual way as $$ax^2+bx+c=0$$ The notes say that, even though that equation can be solved in ...
0
votes
1answer
23 views

Max possible area, of a rectangle shape where one side is a half circle. circumference of 100m

A picture of the shape! I recently took a maths test where one of the questions was just unsolvable for me. I'm going to try to make it as clear as possible, to not create confusion. The question ...
4
votes
4answers
723 views

What is the Difference Between Formulating the Answer via Quadratic Formula and Factoring?

I'm quite eager to learn what is the difference between factoring quadratics (the $(x + a)(x + b)$ method), and using the typical formula (where $x = (-b \pm \sqrt{b^2 - 4ac})/2a$), and in what ...
1
vote
2answers
92 views

Solving a System of Quadratic Equations for Sound Triangulation

I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound. My hypothetical setup includes 3 sensors on a perfectly ...
-1
votes
2answers
29 views

What is the maximum value of $x^2+4xy-y^2$ for all $(x,y)$ satisfying $x^2+y^2 = 1$? [closed]

Does the trick have something to do with the equation of a circle?
0
votes
3answers
67 views

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$ I've tried moving root term to one side and squaring both sides to get a $4$th-degree polynomial and find the roots that way. Is there any easier way of ...
0
votes
0answers
23 views

Calculate quadratic function from given points in 2 dimensions

The quadratic function can be defined as $z = a+b*x+c*y+d*x^2 +e*x*y+f*y^2$ But how to find the 6 coefficients from given truples $(x_i,y_i,z_i)$? I hope that there is a not too difficult solution. ...
2
votes
2answers
32 views

Prove $(X \theta - \vec{y})^T (X \theta - \vec{y}) = \theta^T X^T X \theta - \theta^T X^T \vec{y} - \vec{y}^T X \theta + \vec{y}^T \vec{y}$

I'm studying Machine Learning Stanford's CS229 course and in the lecture note, page number 11, I'm not getting how does step 2 arrive from step 1 above? Prof. Andrew Ng says that it is the expansion ...
1
vote
3answers
31 views

Prove that for $a,p,q \in \Bbb R$ the solutions of: $\frac{1}{x-p} + \frac{1}{x-q} = \frac {1}{a^2}$ are real numbers.

Prove that for $a,p,q \in \Bbb R$ the solutions of: $$\frac{1}{x-p} + \frac{1}{x-q} = \frac {1}{a^2}$$ are real numbers. I tried manipulating the expression, getting rid of the denominators, but i ...
1
vote
2answers
42 views

Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$

Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$ My try: Since $y=x^2+ax+a^2+6a$ is an open upward Parabola, the roots $\alpha,\beta$ should be distinct and satisfy $1 \lt \...
0
votes
2answers
24 views

Quadratic equation solving

How to prove that the value of x from $x^{2}+y^{2}=a^{2}$ and $y=mx+c$ are equal when $c^{2}=a^{2}(m^2+1)$? I tried to equate the x from both variables but can't get it
0
votes
1answer
39 views

quadratic bezier to parabola matrix equation

I'm trying to follow the matrix equation solution presented here: Convert quadratic bezier curve to parabola by @robjohn I'm assuming his solution can be used for any quadratic coordinates. My goal ...