Quadratic Definition and 990 Threads

Hello, Suppose I have a vector space $V$ over $\Bbb R$, a quadratic form $f(x)$ over $V$, some basis of $V$ and a symmetric matrix $A$ corresponding to $f$ in that basis, i.e., $f(x)=x^TAx$. Using, for example, the Lagrange method, I can find a change-of-basis matrix $C$ ($x=Cx'$) such that in...

form quadratic equation $ax^2 +bx+c=0$ in parametric form such that a,b,c are integers in AP and it has got rational roots

If $a,b,c$ are the length of the sides of an scalene triangle, If the equation $x^2+2(a+b+c)x+3\lambda\left(ab+bc+ca\right) = 0$ has real and distinct roots, Then the value of $\lambda$ is given by Options:: (a) $\displaystyle \lambda < \frac{4}{3}\;\;\;\;\; $ (b)$\displaystyle...

How do LQG regulators work? I have read the Matlab page about them, Wikipedia, and a few schools notes on them but it isn't either clear to me or they are not adequately explaining how they work. All see is that we want to control something giving a quadratic cost. Is there are more robust...

form a quadratic equation $ax^2 + bx + c = 0$ such that a,b,c are in AP and it has got integer roots

Hmm I'm having issues with an optional problem for some review for a quiz later tonight. Determine the Quadratic Function for the Graph the points labeled are X intercepts =(-3,0) (5,0); Y-Intercept = (0,-30), Local Min/Vertex = (1,-32) It's the parabola looking problem at the bottom right...

Homework Statement Solve 5x^2 - 16tx + 3t^2 for x without using quadratic formula. I am interested in understanding the technique - I have the answers already: t/5, 3t Homework Equations I have solved the subsequent question which is: tx^2 + (tT - 1)x - T = 0, which seems easier because...

If the quadratic equation $x^2+(2 – \tan \theta)x – (1 + \tan \theta) = 0$ has two integral roots, then sum of all possible values of $\theta$ in the interval $(0, 2\pi)$ is $k\pi$. Find $k$.

I have covered the proofs of the laws of quadratic reciprocity (the Legendre and Jacobi symbols). However this treatment of quadratic residues has been pretty dry. Are there any real life applications of the quadratic residues?

15=35 sin Theta*t -4.9t^2 I have tried solving for t, I do not know how to... please help thank you

Find the maximum of $a+b$, given $a^2-1+b^2-3b=0$.

1. Problem Barkley runs a canoe-rental business on a small river in Pennsylvania. Currently, the business charges $12 per canoe and they average 36 rentals a day. A study shows that for every $.50 increase in rental price, the business can expect to lose two rentals per day. Find the price...

Homework Statement Homework Equations The Attempt at a Solution With a parity operator, Px = -x implies x has odd parity while Px = x implies x has even parity. Things that puzzle me 1. Why is ##[H_0,P] = 0## and ##H_1P = -PH_1##? Is it because ##H_1 \propto z## so ##Pz = -z##? Then...

600*sqrt{1-a^2/400}=578-a^2/2 Shorter and elegant tricks would be welcome!

Does anyone know of an elementary proof of the Law of Quadratic Reciprocity. I am looking for a proof that 1st or 2nd year undergraduate can understand. Searching for a proof which is digestible for a student who is doing a first course in number theory.

1.Homework Statement Derive the quadradic equation, ax^2 + bx + c = 0, using the fallowing method. 1. Divide by a, if a =/ 0. 2. Move the constant to the other side of the equation. 3. Square half the coeffecient of bx and add it to both sides, fixate the negative sign on -c/a. 4. factor the...

Homework Statement Show that the sequence {(p_{n})}^{∞}_{n=0}=10^{-2^{n}} converges quadratically to 0. Homework Equations \stackrel{limit}{_{n→∞}}\frac{|p_{n+1}-p|}{|p_{n}-p|^{α}}=λ where α is order of convergence; α=1 implies linear convergence, α=2 implies quadratic convergence, and so...

the solutions of : $x^2+kx+k=0 " are $ $sin \,\theta \,\,and \,\, cos\, \theta $ please find : $k=?$

Homework Statement Let G:R2\rightarrowR be a C2 function such that G(tx,ty)=t2G(x, y). Show that: 2G(x,y)=(x,y).HG(0,0).(x,y)t The Attempt at a Solution G is C2, so its Taylor expansion is: G(x,y) = G(0,0) + \nablaG(0,0).(x,y) + \frac{1}{2}(x,y).HG(c).(x,y)t, where c lies on...

The question is to classify/describe the following degenerate quadratic surface: x2 - 2xy +2y2 - 2yz + z2 = 0

1.given the equation a(3x2+2x+1)=4x-6x2-4 with the solutions x1 and x2 a) let a=0 without solving the equation calculate x1-3+x2-3 the correct answer is supposed to be -7/2 Homework Equations x1+x2=-b/a x1*x2=c/a The Attempt at a Solution The first thing I was that I put in...

I have a bunch of noisy data points (x,y), and I want to model the data as y = ax2 + bx + c + noise where noise can probably be assumed to be Gaussian, or perhaps uniformly distributed. My data is firmly inside of an interval and I'm only interested in modeling correctly inside of this...

I'm looking for the function y=ax^2+c with the given points (-3,89) (2,39). Thanks

4x^2-y^2+2z^2+4=0 x^2-y^2/4+z^2/2+1=0 -x^2+y^2/4-z^2/2=1 In the xy trace -x^2+y^2/4=1+k^2/2 taking k=0 will yield the hyperbola but what affect will z have on the resulting surface as it tends to +- infinity It appears to me that as z to +- infinity the hyperbola in the xy plane becomes...

z=y^2-x^2 Trying to render these sheets by hand is very difficult for me. I can conceptualize the sheet in general by observing that the trace in z=y^2-k^2 follows the trace of z=k^2-x^2 as z tends to negative infinity. The opposite is also true as z tends to infinity. This information...

Problem: If the curve $y=ax^2+2bx+c$, ($a,b,c \,\in\,\mathbb{R},\,a,b,c \neq 0$) never meet the x-axis, then a,b,c can't be in A)Arithmetic Progression B)Geometric Progression C)Harmonic Progression D)All of these Attempt: Since, the curve never meets the x-axis, we have the condition...

consider the equation: ax^2+bx=c imagine that the constant c is divided into to parts: c=Z+Y where ax^2= Z and bx=Y so basically could we say c is part area and part length?? Sorry if the question is not explained better but that's the best way I came up with to describe it.

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most important in number theory?

Homework Statement Complete the square using the symmetric matrix that defines the given quadratic form: ##x^2 - 4xy + 6xz + 2xt + 4y^2 + 2yz + 4yt + 5z^2 - 6zt - t^2## and write this quadratic as the sum and difference of squares after completing the square using the matrix. The Attempt at...

Homework Statement Solve this system of equations for x and y. v=x+y v^2=x^2+y^2 Homework Equations The quadratic formula: x = (-b +/- sqrt(b^2-4*a*c))/(2*a) The Attempt at a Solution A TA gave the following advice: "Make y the subject of the first equation. Find y2 in terms of v and x...

In equation z=y^2-x^2 when graphing the trace for y^2-x^2=k I see we have y=+-x for k=0 else y=+-sqrt(k+x^2) is there a simple way to graph this

Z=y^2-x^2 Is it always best to substitute k for each and plug in 0, +-1, +-2 to graph each trace.

Z=x^2-y^2 The book is showing the trace for z=0 to be a hyperbola however I see y=x and y=-x

I know for ##ax^2+bx+c=0##, x=\frac{-b^+_-\sqrt{b^2-4ac}}{2a} Using ##x^2-3x-4=0##, we know it is equal to ##(x+1)(x-4)=0##. So ##x=-1## or ##x=4##. but using the formula: x=\frac{-b^+_-\sqrt{b^2-4ac}}{2a}=\frac{3^+_-\sqrt{9+4}}{2}=\frac{3^+_-\sqrt{13}}{2} I cannot get -1 and 4...

On the basis of the eigenvalues of A, classify the quadratic surfaces X'AX+BX+k=0 into ellipsoids, hyperboloids, paraboloids and cylindres. Can somebody help me to solve the problem?

Hi everyone, :) Take a look at the following question. Problem: Determine which of the following quadratic functions \(q_1,\,q_2:\,V\rightarrow\Re\) is positive definite and find a basis of \(V\) where one of \(q_1,\,q_2\) has normal form and the other canonical...

Hi everyone, :) Here's a question with the summary of my method of how to solve it. I would really appreciate if you could go through it and let me know if there are any mistakes with my approach. Also are there any easier methods? Problem: Find an orthogonal transformation that reduces the...

It was given this term: \sqrt{1-(x^2+1)^2} My friend got the solution -x^2,but i think he is not right a about that,cause i believe you should first solve the quadratic binom,which is x^4+2x+1,so my solution is: \sqrt{1-x^4-2x-1} or \sqrt{-x^4-2x} Im wondering who is right now,or how to solve...

This could be seen as a rather "basic" math question, but it is a topic of curiosity for me. I'm currently a senior in high school, taking a pre-ap pre-cal/trig/AP-Calculus double blocked class. I'm absolutely fascinated by mathematics, and something of keen interest to me is the derivation of...

Homework Statement Does every quadratic function have a relative extrema? Homework Equations Quadratic function: ax^2 + bx + c. Aka a polynomial. Polynomials are continuous through all real numbers. The Attempt at a Solution It seems as if all quadratic functions would have...

Hi, I am trying understand if there is a quick way to figure out the impact of the X term in a quadratic equation. For example, by looking at the following equation, I know that it is a parabola (X^2) ; by looking at 24, I know that it is the y-intercept. However, I don't know what is the...

Homework Statement If the quadratic equation ax2+bx+c=0 has equal roots where a, b and c denote the lengths of the sides opposite to vertices A, B and C of a triangle ABC respectively, then find the sum of integers in the range of $$\left(\frac{\sin A}{\sin C}+\frac{\sin C}{\sin...

Homework Statement 4. If x1, x2 are two real number roots of real number coefficient quadratic equation: $$x^2 -2mx + m + 2 =0$$ Solve the following questions: (1) What are the values of m so that x1=x2? (2) What are the...

These 3 equations all describe the same quadratic function. What are the coordinates of the following points on the graph of the function? From which equation is each point most easily determined? y = (x - 5) (x + 1) y = x ^ 2 - 4x - 5 y = (x - 2) ^ 2 - 9 X-intercept, what are the points...

Here are the questions: I have posted a link there to this topic so the OP can see my work.

Homework Statement Question + attempt: Homework Equations The Attempt at a Solution Why is it that when I expand the factored form, I don't get the original equation?

Solve the following quadratic equation. Use factorisation if possible. X2 - 4X - 8 = 0 Normally I wouldn't have trouble factorising a quadratic, but I have just been introduced to a new way to do it and I want to use this way to answer the question. Here's how far I get, then I'm unsure what...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Here are the questions: I have posted a link there to this topic so the OP can see my work.

Homework Statement Solve the equation for x y^2 = 4x x^2 = 4y Homework Equations None The Attempt at a Solution y^2 -4x = x^2 - 4y = 0 I have spent ages re-arranging and substituting in values but I just cannot solve this thing.