Questions on quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

learn more… | top users | synonyms (1)

0
votes
1answer
43 views

Convex optimization: Piece-wise, quadratic objective

This question is about convex optimization with a convex objective function, which is defined piecewise. We have two functions, a concave function A(x) and a strictly convex, increasing function B(x), ...
1
vote
2answers
164 views

$a,b,c$ are the sides and $A,B,C$ are the angles of a triangle. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then,

$a,b,c$ are the sides of a $\triangle ABC$ and $A,B,C$ are the respective angles. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then $\sin^2 \bigl(\frac{A}{2}\bigr), \sin^2 \bigl(...
0
votes
0answers
15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: $$m_n(x)=f(x_n)+\...
13
votes
1answer
313 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...
-2
votes
2answers
231 views

solving $y=4/x+\sqrt{x+0.2−5x}$ [closed]

it's actually $y=\frac{4}{x}+\sqrt{x+0.2-5x}$ (see algebra problem) $$y=\frac{4}{x}+\sqrt{x+0.2-5x}$$ if $x=\frac45$ what is y?
2
votes
1answer
48 views

Weird property of quadratic equations

Today, we learnt about the quadratic formula, and I noticed a strange property of (seemingly) all quadratic equations. If we call the two solutions of any arbitrary quadratic equation $x_1$ and $x_2$: ...
4
votes
3answers
43 views

Possible number of terms in an Arithmetic Progression

The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is $a)$ $3$ $b)$ $4$ $c)$ $...
1
vote
0answers
33 views

How to constrain the linear least squares fit of a quadratic polynomial with known constraints

How to constrain this fit I have some function , $f(x) = a x^2+b x+c$ , with the constraints $a<0$ and $c = \frac{b^2}{4a}+\frac{1}{2}ln(\frac{-a}{\pi})$ I have measured $f(x)$ for some $x$. Can ...
1
vote
4answers
554 views

How do the roots change

How do the roots of the quadratic equation $ax^2+bx+c=0$ change when $b$ and $c$ retain constant values and $a$ tends to zero? ($b\neq 0$)
0
votes
2answers
51 views

What is the solution set of $ \displaystyle\log _x\left({x+5\over 1-3x}\right)>0$?

What is the solution set of $ \displaystyle\log _x\left({x+5\over 1-3x}\right)>0$? I'm getting $0 < x < 1/3$ but that is wrong answer.
1
vote
0answers
28 views

Solve quadratic vector equation, with variable hidden inside scalar

Let $\vec{f}$ $m\times 1$ unknown vector, given $n\times 1$ vector $\vec{F}$, $n\times m$ matrix A ($n<m$), nonzero vector $\vec{v}$ from the nullspace of A ($Av=0$), non-invertible symmetric ...
-1
votes
2answers
66 views

Correct proof of a theorm [closed]

Hi Can someone please give me an idea where to start with this Thanks Steve
2
votes
1answer
43 views

What curve is traced out by the vertex of the parabola $ax^2+bx+c$ as $b$ varies?

Consider the parabola $\rho$ given be the equation $y = ax^2 + bx + c$. Recall that varying $a$ in this equation stretches/squashes $\rho$ and that varying $c$ shifts $\rho$ vertically. The change in ...
0
votes
1answer
31 views

How to find the integral value of $a$ for which $f(x) = x^2 - 6ax + 3 - 2a + 9a^2$ is surjective

Let $f:\mathbb{R} \to [1, \infty)$ be defined by $f(x)=x^2-6ax+3-2a+9a^2$. The integral value of $a$ for which $f(x)$ is surjective is equal to I tried putting $f(x)=1$. Is this the right approach?
1
vote
1answer
26 views

Composition of three quadratic functions

Is it possible to find three quadratic functions $f(x),g(x)$ and $h(x)$ such that $f(g(h(x)))$ has $-6,-5,-4,-2,1,3,4,5$ as its roots? I understand that the composition of three quadratic functions ...
0
votes
1answer
31 views

Find $f(\mathbb D)$

If $f(x)=\frac{\sqrt{x}-x}{\sqrt{x}+2}$ and $x\in\mathbb D=[0,\infty]$ find $$f(\mathbb D)$$. I've tried to solve equation $y=f(x)$ and stopped to $x+(y-1)\sqrt{x}+2y=0$.
1
vote
3answers
40 views

Why does the width of the graph of a parabola depend only on $a$, not $b$?

Lets assume a quadratic function $y = ax^2 + bx +c$. My book says how wide or narrow the graph is depends on the size of $|a|$ My question is why doesn't it depend on $b$ also? If you, say, increase ...
4
votes
3answers
99 views

Find $a^{100}+b^{100}+ab$

$a$ and $b$ are the roots of the equation $x^2+x+1=0$. Then what is the value of $a^{100}+b^{100}+ab$? Here's what I found out: $$a+b=-1$$ $$ab=1$$ but how to use this to find that I don't know! ...
3
votes
1answer
89 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
6
votes
8answers
951 views

Solving the following trigonometric equation: $\sin x + \cos x = \frac{1}{3} $

I have to solve the following equation: $$\sin x + \cos x = \dfrac{1}{3} $$ I use the following substitution: $$\sin^2 x + \cos^2 x = 1 \longrightarrow \sin x = \sqrt{1-\cos^2 x}$$ And by ...
2
votes
1answer
22 views

How to determine the number of free variables in a nonlinear system?

Consider the system of equations $$ \begin{align} a_1 b_2 &= c_1 \\ a_1 b_3 &= c_2 \\ a_2 b_1 &= c_3 \\ a_2 b_3 &= c_4 \\ a_3 b_1 &= c_5 \\ a_3 b_2 &= c_6 \\ \end{align} $$ ...
0
votes
2answers
23 views

Invertability of quadratic function

There is a proof in my lin alg book that shows that if T is a self adjoint operator, $T^2+Tb+c$ is invertible when $b^2 < 4c$?. I understand this proof. Why is it that the above holds, and yet the ...
0
votes
2answers
52 views

Simply factoring a quadratic equation

On pp 255 - 256 (footnote 7) of "Love & Math", Edward Frenkel states that we can factor a quadratic in terms of its solutions $x_1$ and $x_2$ as: $ax^2 + bx + c = a(x - x_1)(x - x_2)$ Where does ...
1
vote
2answers
51 views

Roots of a quadratic equation

If $ a $ , $b$ are the roots of $ x^2+x+1=0 $ , then the equation whose roots are $a^k $ and $b ^k$ where k is a positive integer not divisible by 3 is $a)$ $x^2 - x + 1 = 0$ $b)$ $x^2 + x+1 = 0$ $...
5
votes
2answers
69 views

What is the range of $f :R → R$, and $f(x) = x^2 + 6x − 8$

I have this discrete math question I have done completing the square but not sure how to continue. May I get some guide? Thanks! What is the range of $f :R → R$, and $f(x) = x^2 + 6x − 8$ $f(x)=x^2+...
1
vote
1answer
60 views

Finding integers satisfying an equation.

Suppose $p$ is a prime greater than $3$. Find all pairs of integers $(a,b)$ satisfying the equation $$a^2+3ab+2p(a+b)+p^2=0$$ A good way to start (probably) was to complete the square, $$p^2+2p(a+b)...
3
votes
6answers
159 views

Find the roots of quadratic polynomial given one root of another quadratic polynomial?

if $a,b,c$ are Real numbers and $1$ is a root of equation $ax^2+bx+c=0$ then curve $y = 4ax^2+3bx+2c$ , (a is not zero) intersects $x$ axis at how many points? I get a relation $a+b+c = 0$ I tried ...
0
votes
1answer
34 views

Determining the number of roots

Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:- $$ax + by + c = 0 \tag 1$$ $$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$ What are the conditions that can be ...
1
vote
0answers
45 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem (...
0
votes
1answer
25 views

Linear-Equation - What is the error in this equation?

I want to know if this equation is wrong? \begin{align} 4(x - 3) + 2(x - 1) - 3 &< 10\\ 4x-12 + 2x-1-3 &< 10\\ 4x - 2x &< 10 + 12 + 1 + 3\\ 2x &< 26\\ ...
0
votes
2answers
206 views

How to use the property $\sqrt{a}\sqrt{b}=\sqrt{ab}$ with caring that it not always holds?

I was solving a question from my book. If $\alpha$, $\beta$ are the roots of $pt^2+qt+q=0$, $p \neq 0$ and $q \neq 0$ then show that, $\sqrt{\frac{q}{p}}+\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\...
1
vote
1answer
22 views

Troubleshooting Textbook: Using Generating Functions for Non-Homogenous Recurrence

I am learning about Generating Functions to solve non-homogenous linear recurrences, and I can't seem to get the right solution, no matter what I do. Also, the example I have to work off of in the ...
0
votes
1answer
28 views

Forming Quadratic equation from roots

I think I just need some background. I've got the following quadratic equation: $$ 1 - x - 2x^2 = (1-2x)(1 + x) $$ But if I solve it with the quadratic equation, I get the roots: $$ \frac{1 + \...
1
vote
2answers
32 views

Simplifying a solution to a quadratic equation

I am solving: $(\sigma_A^2 - 2\rho\sigma_A\sigma_B +\sigma_B^2)x^2 +2(\rho\sigma_A\sigma_B - \sigma_B^2)x +\sigma_B^2 = 0$ I need to show that a real $x$ exists if and only if $\rho = \pm 1$ Using ...
1
vote
5answers
31 views

Existence of rational roots in a quadratic equation

Consider the quadratic equation $(a+c-b)x^2 + 2cx + b+c-a = 0 $ , where a,b,c are distinct real numbers and a+c-b is not equal to 0. Suppose that both the roots of the equation are rational . Then a) ...
0
votes
1answer
128 views

Existence of a common root among two quadratic equation

The equations $x^2 + x + a = 0$ and $x^2 + ax+ 1 = 0$ a) Cannot have a common real root for any value of a b) have common real root for exactly one value of a c) have a common real root for exactly ...
1
vote
1answer
54 views

What are the solutions for $2^x=x^2$? [duplicate]

What are the solutions for $2^x=x^2$? I noticed there were 2 roots: $2,4$. Are there any other roots, and how do you calculate them?
3
votes
1answer
75 views

Trying to prove $c^3a^2+(9c^2-b^2)a+(27c-10b)=0$ has no positive integer solutions

I'm trying to prove (or, I suppose, disprove) the following claim, in either version. Conjecture (Strong Version): There are no positive integers $a,b,c$ such that $$c^3a^2+(9c^2-b^2)a+(27c-10b)=0.$$ ...
2
votes
2answers
46 views

If the range of the function $f(x)=\frac{x^2+ax+b}{x^2+2x+3}$ is $[-5,4],a,b\in N$,then find the value of $a^2+b^2.$

If the range of the function $f(x)=\frac{x^2+ax+b}{x^2+2x+3}$ is $[-5,4],a,b\in N$,then find the value of $a^2+b^2.$ Let $y=\frac{x^2+ax+b}{x^2+2x+3}$ $$x^2y+2xy+3y=x^2+ax+b$$ $$x^2(y-1)+x(2y-a)+3y-...
1
vote
2answers
47 views

Let $f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}$ is defined for all real $x$,then find the number of possible ordered pairs $(a,b),$

Let $f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}$ is defined for all real $x$,then find the number of possible ordered pairs $(a,b),$ where $a,b$ are both integers. As $f(x)$ is defined for all real $x$,...
3
votes
4answers
79 views

Can every parabola be written in the form of a quadratic $y=ax^2+bx+c$ or $x=dy^2+ey+f$?

I understand that the graph of any equation of the form $y=ax^2+bx+c$ is a parabola (please correct me if I am mistaken). My question is about the converse: Can every parabola be written in the form ...
1
vote
2answers
38 views

How to find the unknown in this log inequality??

Find all values of the parameter a $\in\Bbb R$ for which the following inequality is valid for all x $\in\Bbb R$. $$ 1+\log_5(x^2+1)\ge \log_5(ax^2+4x+a) $$ I'm lost when I got to this stage: $ 5x^...
0
votes
1answer
42 views

If the roots of $ax^2+bx+c=0$ are of the form $\frac{m}{m-1}$ and $\frac{m+1}{m}$ then find..

Problem : If the roots of $ax^2+bx+c=0$ are of the form $\frac{m}{m-1}$ and $\frac{m+1}{m}$ then find the value of $(a+b+c)^2$ My approach : Let $\alpha, \beta$ are the two roots of the given ...
1
vote
2answers
32 views

Expansion and factorisation

I have a little problems with a few questions here and I need help.. Thanks ... Factorise completely $$9x^4 - 4x^2 - 9x^2y^2 + 4y^2 $$ My workings .. $$ (3x^2+2x)(3x^2-2x) - y^2 (9x^2-4) = (3x^...
0
votes
1answer
22 views

From the graph of a quadratic equation, find the range of values of $x$.

The equation of the graph is $$ y = -x^2 + 9x - 18 $$ From the graph sketched, find the range of values of $x$ for which $x^2 + 18 > 9x$. Workings $$ y = -(x-3)(x-6) $$ I'm not sure what is the ...
1
vote
3answers
53 views

If $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$.

If $a,l\neq0$ , $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$. How can we exactly say (how to prove) that $p(x)$ is a quadratic ? What methods can be used to ...
0
votes
1answer
36 views

Does any quadratic function in the form $an^2 + bn + c$ equal $\Theta(n^2)$ in asymptotic notation?

On a Khan Academy post (see here) about Big-$\Theta$ notation, the author attempted to convert the quadratic function $6n^2 + 100n + 300$ to asymptotic notation. They started by dropping the $n^2$ ...
0
votes
1answer
38 views

solving an equation $x^x= c$ [duplicate]

I would like to find a solution $x$ for $x^x = c$ where $c$ is a positive constant. Firstly I'm looking for an approximative solution when $c$ tends to infinity. Thank you in advance
0
votes
2answers
23 views

Quadratic equations - Fastest way to find the value of d

$\frac{2d^2-d-10}{d^2+7d+10} = \frac{d^2-4d+3}{d^2+2d-15}$ What is the optimal solution for finding the value of d?
0
votes
4answers
62 views

Find the real values of $P$ for which $f(x)=P$ has exactly one solution.

$$f(x)=(x-1)^2(x-2)+1$$ Find the real values of $P$ for which $f(x)=P$ has exactly one solution. Hi, I'm a little bit confused with this question.I don't know how I should start. I think I need ...