In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
4
x
2
+
2
x
y
−
3
y
2
{\displaystyle 4x^{2}+2xy-3y^{2}}
is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If
K
=
R
{\displaystyle K=\mathbb {R} }
, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form).
Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.
Homework Statement
Classify the quadratic surface:
2x^2 + 4y^2 - 5z^2 + 3xy - 2xz + 4yz = 2
Homework Equations
A quadratic form is a second degree polynomial equation in three variables which has the form
F(x,y,z) = ax^2 + by^2 + cz^2 + 2dxy + 2exz + 2fyz
where coefficients a...
Homework Statement
Let the quadratic form F(x,y,z) be given as
F(x,y,z) = 2x^2 + 3y^2 + 5z^2 - xy -xz - yz.
Find the transitional matrix that would transform this form to a diagonal form.
Homework Equations
A quadratic form is a second degree polynomial equation in three...
An object accelerates at the rate of 1.30 m/s^2 over a displacement of 11.8m, reaching a final velocity of 7.00 m/s. For what length of time was it accelerating? Explain the answers.
NOTE: Actually, when I first was about to post this, I finally had gotten the answer I needed. However, I cannot...
Okay, this is something that's been bugging me for a while. A lot of the polynomials with higher powers can't be solved using algebraic methods and must be estimated. So my idea was to take these answers and find a way to convert them into an exact form, such as quadratic irrationals/surds...
Let P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}, and assume that Q is positive definite, P^ - is positive semidefinite whence (MP^ - M^T + Q)^{ - 1} exists (where T denotes transpose).
In what sense does K = P^ - M^T(MP^ - M^T + Q)^{ - 1} minimize the quadratic...
Hello PF,
Ok so here is my problem. I have an array of data points which are the chi^2 values for a fit I have performed. I wish to calculate the 1-sigma error for this fit, so I figure I could find the lowest chi^2 value, and the two points either side that value, and fit a quadratic...
Homework Statement
It Costs a bus company $225 to run a minibus on a ski trip, plus $30 per passenger. the bus has seating for 22 passengers, and the company charges $60 per fare if the bus is full. For each empty seat, the company has to increase ticket price by $5. How many empty seats...
Homework Statement
A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum...
Homework Statement
A light bulb is wired in series with a 144Ohm resistor, and they are connected across a 120.0-V source. The power delivered to the light bulb is 23.4W. What is ther resistance of the light bulb? Note that there are two possible answers.
Homework Equations
P = I^2R...
[b]1. A farmer grows orange trees. He has 20 trees/hectare with 300 oranges/tree. When adding another tree to a hectare, the amount of oranges decreases by ten. What is the max yield of trees. (They want to know how many trees they should have in a hectare to maximize their orange production.)...
[b]1. Suppose y = bx^2 +cx + a
a.) in terms of a, b, and c, what values of x make y=0?
b.) if a=3.1, b= -2.2 and c=-4.3 evaluate those solutions to 3 significant digits:
Homework Equations
[b]3. I am not really sure how to solve for what the question is asking for in part...
Homework Statement
Identify the vertex and x-intercept(s) of the quadratic function algebraically.
Homework Equations
f(x) = 1/2x^2 - 4
1/2 Is the coefficent if its not clear.
The Attempt at a Solution
Ok I do not need help finding the vertex I have already got that, I...
Homework Statement
I have a probability distribution function (PDF) going fromi 0 to a on the y-axis and 0 to 30 on the x-axis. The function describing the PDF is the quadratic equation f(x) = a+bx+cx^2
I have to find a, b, and c.
Homework Equations
The Attempt at a Solution...
Hi all,
I need to construct the Casimir op. of group SU(3).
I have these relations;
T2=\sum C_{i}_{j}T_{i}T_{j} i,j=1,2...,8 ...eq1
[Ti , Tj]= \sum f_{i,j,k} T_{k} ...eq2
[T2 , Ti]=[\sum C_{i}_{j}T_{i}T_{j} , Ts]=\sum C_{i}_{j}T_{i}[T_{j}, T_{s}] + \sum C_{i}_{j}[T_{i}...
Homework Statement
Y(s)=(s^2+s-6)/(s^2-2s+5)
The Attempt at a Solution
I tried splitting up the bottom into the form (s-1)^2+4, but I can't get the top to factor correctly to make that work. I'm not sure where to go from here, since I can't do partial fractions with the top and bottom...
k so i used
x-xo= Vo t + 1/2 at^2
plugged in all my values
then moved everything to one side and used the quadratic formula
here is my question.
when i solve for it i get
t=40.28 and t= 40.58
So which t is my time?
Q- A number is a quadratic residue modulo m if it takes the form x^{2} mod m
for some integer x. List the quadratic residues modulo 3, 4, 5, and 7. What
patterns, if any, do you notice?
modulo 7
0^2=0, 1^{2}=1, 2^{2}=4, 3^{2}=2, 4^{2}=2, 5^{2}=4, 6^{2}=1
modulo 5
0^2=0, 1^{2}=1, 2^{2}=4...
Quadratic equation without numbers??
How would one go about solving an equation such as B2X2+2ABCX-ABC2 for the value of x that equals zero?
All of the letters are constants except for X, so would this be a quadratic equation problem.?
The solution in the book is somehow...
1. Consider the quadratic equation ax^2+px+aq+q=0 where a does not equal to zero & p and q are constants . It is known that one of the roots of the quadratic equation is always 1 regardless of the value of a. Prove that p+q=0
[b]3. I have tried to factorize it and it became x(ax+p)+q(a+1)=0...
Homework Statement
At the end of a much longer problem, I'm asked to find a second height that will satisfy a formula found for a height in the first part of the problem where:Homework Equations
h1(h2-h1)=h3(h2-h3)The Attempt at a Solution
I know the answer I should get: h3=h2-h1
But I cannot...
Homework Statement
2 cos x + tan x = sec x
Homework Equations
I can move terms around with identities, but I'm stuck with the partially solved equation below. I don't know how to solve a quadratic with weird terms. I got really far. But how do I show sin x?
sin x = (-1 +- sqrt...
Quadratic function...help!
Hi
I have an exam (next week) on quadratic functions! (10th grade) Any online websites where I can practice?I need problematic situations(there's a problem - you analyze - calculate stuff,etc)... By the way, I'm studying in a FRENCH HIGH SCHOOL, therefore the...
I am currently studying the Mandelbrot set, and have a question about one of the statements on http://mathworld.wolfram.com/QuadraticMap.html" .
It says the recurrence for the Mandelbrot set "is not in general solvable in closed form." What does this mean?
For an odd prime number p let Fp be the field with p elements, ie. the integers {0...,p-1} with addition and multiplication defined modulo p. How many quadratic forms are there on the vector space Fp^n
I don even know how to start this question
Hi there!
I need a little help from your side. Could you give me an example of a quadratic number in Z3
there is a lemma, that says there are exactly two quadratic numbers there, but I somehow cannot figue out how to find them :(
I know that there are 3 equivalence classes there: [0]...
Homework Statement
I just need some guidance regarding how to work out the value of x in the quadratic equation. I don't want you guys to do the work for me, I just want to know how to figure it out.
Thanks
Homework Equations
The Attempt at a Solution
Homework Statement
A particle that can move in one dimension and that is in a stationary state, is bound by a potential V(x) = (1/2)kx^2. The wave function is \Psi(x,t) = \psi(x)exp(-iEt/\hbar)
We look at a state in which \psi(x) = Aexp(-x^2/2a^2a^2), where a is a constant and A is the...
Homework Statement
In the attached diagram, the displacement of a follower (y) changes as the angle of the cam changes (\vartheta). The follwer has a smooth velocity over the entire path. Therefore, the velocity must match on both sides of the four points noted on the graph.
If we assume...
Here is what I know:
1) All quadratic curves of 2 variables correspond to a conic section.
ax^2 + 2bxy +cy^2 + 2dx + 2fy + g = 0
a, b, c are not all 0
2) The definitions of parabola (in terms of a directrix and focus), ellipse (in terms of 2 foci), hyperbola (in terms of directrix and...
P = 87x - 700 -x^2
1. Draw a graph of P against x and estimate the maximum value of P and also calculate the coordinates of the stationary point and verify if it is maximum.
I am not very good at graph drawing or stationary points... any help people?
Hey
I'm wondering if anyone can give me a proof to the quadratic equation solution which is not the simple add here multiply that proof
(I just recall some cooler proof, which I think requires more "complicated" mathematics than adding and multiplying, though I can't remember what it was)
Homework Statement
lxl <2
lx+2l
The question is asking to solve this
Homework Equations
The Attempt at a Solution
Ive tried bringint the 2 over which leads me to l-x-4l over lx +2l < 0 but then the absolute value confuses the heck out of me on where to go...
SOLVED
Homework Statement
While going through a question, I came across a function f(x)=5^{x}+5^{-x}
I saw its graph through a software. It was a parabola with minimum value 2.
Now a question arises in my mind.
Every function of the type g(x)=ax^{2}+bx+c is a parabola.
Can I assume...
I have to find a system of linear equations to determine a quadratic polynomial
p(x) = ax2+bx+c
Given the conditions:
p(1) = f(1)
p'(1) = f'(1)
p''(1) = f''(1)
where f(x) = xex-1
I know:
p(x) = ax2+bx+c
p'(x) = 2ax +b
p''(x) = 2x
f(x) = xex-1
f'(x) = ex-1 +...
I just happened to read two papers that pretend that the quadratic divergence of the Higgs mass is not a problem.
The first is "Vacuum energy: Quantum Hydrodynamics vs Quantum Gravity" http://arxiv.org/abs/gr-qc/0505104 (Update: this is now the correct paper from arxiv) where Volovik says that...
I'm fine solving DE with a reducible quadratic, but what happens when the quadratic is irreducible, how would I go about solving the DE? So for example,
\frac{dP}{dt}=-P^{2}+10P-26
Thanks in advance.
I got into college as a mature student and I'm doing well in all subjects except for math. I'm struggling a good bit because I never finished high school math and there's a lot of stuff they assume everyone knows but I don't. I have to study it at home so I can keep up. Its mainly quadratic...
HI Can anybody tell me how to calculate a PDF of y, where y is a function of x, such that
y = a X*X + bX + C (i.e. a quadratic equation), and X follows the Normal Distribution X ~N(0, sigma)
Help anybody?
Thanks
Homework Statement
Consider ax''+bx'+cx=0 for b^2-4ac=0 with k= -b/(2a).
Show x=e^(kt) is a solution.
The Attempt at a Solution
x=e^(kt)
x'=(-b/2a)e^((-b/2a)t)
x''=(-b^2/4a^2)e^((-b/2a)t)
Then i plugged these into ax''+bx'+cx=0 and simplified to get e^((-b/2a)t)(-((b^2)/2)+c)=0...
here is my task: (x³-4x²+5x-2):(x-1)=
We did such task in school but I didn't got it ..
Here is the task that we did in school
(x³-3x²-6x+8) :(x-1)= x²- 2x-8 but I don`t get it why x² is the result and why 3x² ??
Can someone help me? Thank you in advance
I have a problem with this question:
let q : V -> R is quadratic form and suppose T = {v|q(v) \geq 0} is subspace of V. prove that q is indefinite.
Thanks in advance.
Homework Statement
A gun is fired straight up. Assume a quadratic drag force.
Show
(upward motion) v^2 = Ae^{-2kx} - \frac{g}{k}
(downward motion) v^2 = g/k - Be^{2kx}
Homework Equations
The Attempt at a Solution
I can easily solve the integrals unfortunately I am having difficulty setting...
If one plots the graph of
y = x2-10 x+24
which is a parabola, one observes that it is above the x axis, i.e. y > 0, for two intervals of the real line. (The union of these intervals constitute the set of solutions of the quadratic inequality at the beginning of this question.)
Find...
I had a hard time completing the square of some problems. This is what I did with them, I would appreciate if someone could show me the mistake that I am doing and the right process in order to solve these problems.
Also, here there is an article about completing the square for anyone that...