Quadratic Definition and 965 Threads

Hi all, There is a Hamiltonian in terms of "a" and "a^{dagger}"bosonic operators H=ω*(a^{dagger}a+1/2)+alpha*a^2+β*a^{dagger}^2 and ω, alpha and β are real constants and its energy is E=(n+1/2)*epsilon where epsilon is ω^2-4*alpha*β. Now, I tried to find this energy but I couldn't. Would you...

how do you find the quadratic equation given (-3,0) (3,0) and (0,-4) thus y=k(x-3)(x+3) for the zero's but how do you find k or any other better method of finding the equation thanks much(Cool)

I am trying to figure out how the coeffecients in the general second degree equation transform the function as they change through a range of values. I think that d and e are the proportion of x and y are horizontal and vertical shifts, but less sure about a, b, and c. Here is the equation as...

Without the classic method of "completing the square" and factorising?

Homework Statement For any quadratic polynomial ax2+bx+c having zeros β and α Prove that β + α = -b/a and αβ = c/a. Homework Equations The Attempt at a Solution I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a. It goes like this. If α and β are the...

Homework Statement 1. Neglecting the height of release, a ball is thrown vertically upwards at 20 m/s, find: d) time (s) that the projectile is at height 10.5 m. V1 = 20m/s a = -9.81m/s^2 d = 10.5m Homework Equations d = v1*t + 1/2 a(t^2) The Attempt at a Solution...

Homework Statement Find all the values of 'a', so that exactly one root of the equation x2-2ax+a2-1=0, lies between the numbers 2 and 4, and no root is either equal to 2 or equal to 4.Homework Equations The Attempt at a Solution Let f(x)=x2-2ax+a2-1 I tried to visualize the question by graph...

Homework Statement For a constant k, consider the function f(x) = x2 + kx + k2 - 2k - 4 Find the range of k for which f(x) = 0 has one solution between 0 and 1 and the other solution between 1 and 2Homework Equations quadratic formula discriminantThe Attempt at a Solution x2 + kx + k2 - 2k - 4...

Hello, I just want to clear up a confusion. Is f(x,y) = a + bx + cy + dxy a quadratic polynomial? where x and y are variables and a,b,c,d are constants.

Hey, I was going through some mechanics notes and came across this quadratic equation to solve for x. In my notes it is supposed to equal 3 however I do not get that result. I am not sure if I am simplifying it wrong or what. I am ending up with 2.12 for the positive value. Please let me...

http://www.screencatch.com/screenshots/13354664404923.jpg

Homework Statement Find all solutions to z^2 + 4conjugate[z] + 4 = 0 where z is a complex number. Homework Equations Alternate form: 4conjugate[z] + z^2 = -4 The Attempt at a Solution I have tried solving this solution using the quadratic formula. However, √b^2 - 4ac = √16 - 4x1x4 = 0...

Homework Statement A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s2. For how much time and what distance does the...

solve √(5x-1) - √x=1 eq. 1 transpose one of the radicals: √(5x-1)=√x+1 Square: 5x-1=x+2√x+1 Collect terms: 4x-2=2√x or 2x-1=√x Square: 4x^2-4x+1=x, x=1/4 , 1. x=1/4 is not the root by substitution to equation 1. ... My question what makes x =1/4 is not the root, only x=1 is the root.

Homework Statement The following is part of a solution to a problem about finding equiblirum points of a differential equation and sketching its bifurcation diagram http://img32.imageshack.us/img32/9194/63226925.jpg How did they get y= -1 \pm \sqrt{1- \mu}? Homework Equations The...

[FONT=Tahoma]if $f(x)$ be a continuous and assumes only rational values so that $ f(2010) =1. $ then roots of the equation $f(1)x^2 + 2f(2)x + 3f(3) =0$ are

I have to find the vertex, axis, domain, & range of this quadratic equation: f(x)= -1/2(x+1)^2 -3 I can't remember how to do the vertex. Is it y= -b/2a? Also, I tried factoring the equation, but I think I messed up. I ended up with: f(x)= .25x^2 +.5x-2.75 Thanks

I'm sorry, I just realized I put this in the wrong subsection. While I figure out how to fix that, please have a look anyway. __ Homework Statement Given x \inℝ And s =\frac{4(x^{2}) + 3}{2x-1} Prove that s^{2} -4s - 12 ≥ 0 Homework Equations The discriminant Δ, (in order for which to be...

Homework Statement Let the roots of x2 + x + 1 = 0 be \alpha and \beta, form a quadratic equation with roots: (i) \alpha2, \beta2; and (ii) \frac{1}{\alpha}, \frac{1}{\beta}.The attempt at a solution sum of roots = \alpha + \beta = -1 product of roots = \alpha\beta = 1 (i) sum of roots =...

Homework Statement Prove that if p is prime, then if A is a quadratic nonresidue mod p and B is also a quadratic nonresidue mod p, then AB is a quadratic residue mod p. Homework Equations A is a nonresidue means A = x^2 (mod p) has no solutions The Attempt at a Solution I already...

Hi group, I'm currently trying to understand a physicists written paper on ecological models. In there, they used the term "quadratic potential" when comparing spatial diffusion with a spring system (see attachment 1). After searching online for this term, I found nothing directly relevant to...

Homework Statement IMPLICIT NONE REAL :: A, B, C, DISCR, X1, X2, x1i, x2i CHARACTER(3)::ANS,ANS1 ! Reads the coefficients for the quadratic equation 33 WRITE(*,*)")"Please enter a *REAL*NUMBER* coefficient ( A )." READ(*,*)A WRITE(*,*)"Please...

2x^2+9x-5 here is the quadratic expression that I am trying to factor. Is there a way to factor this easily? I am getting confused on the signs. the book says the factor are: (2x-1)(x+5) but (-1)(2)+5=3 but why not (2x+5)(x-1)? and 5(2)=10-1=9 which looks like the correct answer...

Homework Statement Canada Post will deliver parcels only if they are less than a certain maximum size: the combined length and girth cannot exceed 297 cm (Girth is the total distance around the cross-section of the parcel). Canada Post delivers a crate with the smallest SA to your house. What...

If a is a quadratic nonresidue of the odd primes p and q, then is the congruence x^2 \equiv a (\text{mod } pq) solvable? Obviously, we want to evaluate \left( \frac{a}{pq} \right) . I factored a into its prime factors and used the law of QR and Euler's Criterion to get rid of the legendre...

Hi all. I keep coming across the term "quadratic gravity" in various GR papers and don't entirely understand the concept as it doesn't seem to be explained anywhere. It seems to appear most frequently in the context of pp-waves. So I was hoping somebody could enlighten me as to what its all...

I need to prove that a be a odd integer that congruence X^2\equiva mod 2 is always solvable with exactly one incongruent solution modulo 2. this question is linked with (b) let a be an odd integer. Prove that the congruence X^2\equiva mod 4 is solvable iff a\equiv1 mod 4. in this case ,prove...

number theory -- quadratic residues Homework Statement find all incongruent solutions of each quadratic congruence below. X^2\equiv23 mod 77 Homework Equations X^2\equiv11 mod 39 The Attempt at a Solution it is suffices to X^2\equiv23 mod 7, andX^2\equiv23 mod 11, then how to do next?

1. The problem: Find the value of \lambda for which the sum of squares of the roots of the equation: 2x^2 + (2\lambda +4)x^2-(1+ \lambda) = 0 has minimum value. 2. Homework Equations \alpha + \beta = -b/a \alpha\beta = c/a where a,b,c are coefficients of x^2 , x and constant...

Homework Statement The profit, P, of a video company, in thousands of dollars is given by P=-5x^2+550x-5000, where x is the amount spent on advertising in thousands of dollars. a) Determine the amount spent on advertising that will result in a profit of $0. Homework Equations...

Homework Statement For each of the following, identify the base function and describe the transformation(s): f(x)=-4(3x)^2 + 5 Homework Equations f(x) = -4(3x)^2 + 5 The Attempt at a Solution Alright so my attempt at figuring out the requested answers are: Base function = x^2...

Homework Statement By expanding (x-y)^2, prove that x^2 +y^2 ≥ 2xy for all real numbers x & y. Homework Equations The Attempt at a Solution expanding (x-y)^2 x^2 - 2xy + y^2= 0 Hence, x^2 + y^2 = 2xy But where does the ≥ come into it? and why? when you put values in...

Homework Statement I'm studying for the GMAT and the study materials do not include how to solve this equation. I need to know how to solve it without a calculator. Homework Equations If 2^x - 2^x-2 = 3(2^13), what is the value of X? The Attempt at a Solution The answer key says...

Homework Statement Rotation of axes to Eliminate the xy-term in a quadratic equation 3x^2 - 10xy + 3y^2 + 16 √2x - 32 = 0 2. The attempt at a solution I do not know the method that should be taken to solve this sort of question. That is what I am looking for. If anyone can point me...

Homework Statement Homework Equations distance = rate * time The Attempt at a Solution So the speed of the ball I have as: [t = time] 5 + 32t Which I believe makes sense because the ball is initially traveling @ 5 ft/sec, and after 1 second, the ball is traveling 37...

Homework Statement Hello! z^2 - (7+2i)z + (12+8i) = 0 The Attempt at a Solution I used the quadratic equation and got: z = (7+2i)/2 +- sqrt( -3/4 - i) r = 5/4, and the argument is Theta = arctan(-1/-3/4) = 0.927 925 - Pi (since x < 0 and y < 0, right?) Then, I know that z =...

Homework Statement The following equation defines a rational function.Find its Range f(x) = x2/(x-1) The attempt at a solution Let y = x2/(x-1) =>y(x-1) = x2 =>xy-y = x2 =>x2-xy+y = 0 comparing with the general form of quadratic equation ax2+bx+c = 0, a=1, b=-y, c=y...

A quadratic equation has the form y = ax^2 + bx + c. However, if c = 0, then y = ax^2 + bx. Is it still called a quadratic equation? And if b = 0 so that y = ax^2, is it still given the title of quadratic equation? I would guess yes since it still has a power of 2 and is a parabola. Is...

Homework Statement Motorbike crossing a river using a ramp. River is 40m wide, the bank on the opposite side is 15m lower. The ramp angle is 53 degress. What is the speed needed at take off? Homework Equations I am using s = ut for the horizontal and s = ut x 1/2 at2 I think because the bank...

I'm told that factoring is an important skill in calculus so I am avoiding using the quadratic formula. But for quadratic equations with large coefficients to factor, is there a better/faster way rather than guessing and checking every single combination?

Hello there, currently I am trying to solve a least squares problem of the following form: min_{M} ||Y - M*X||^2 where M is a 3x3 matrix and Y and X are 3xN matrices. However, the matrix M is of a special form. It is a rank 1 matrix which satisfies M*M = 0_{3x3} and the trace of M is zero...

Homework Statement Get an inverted quadratic equation (-x^2) with maximum y=100 and roots x=0 and x=1/120. Homework Equations The Attempt at a Solution I tried changing the several terms of the quadratic equation, used the quadratic formula setting c=100, and x=0 with the...

I am reading Dummit and Foote Chapter 7. D&F use a quadratic field as an example of a ring. I am trying to get a good understanding of this ring. D&F define a quadratic field as follows: Let D be a rational number that is not a perfect square in and define \mathbb{Q} ( \sqrt D ) =...

Homework Statement Hey guys, recently, I struggled to solve this equation. I need to find θ1 and θ2 by using this system of equations: θ1+θ2=γ θ1*θ2=β After letting θ2=β/θ1, I plugged this into the first equation. I got: θ12-γθ1+β=0 At this point I get a very unsexy θ1 result by using...

Hi, Everyone: I have a job interview tomorrow where I must give a 10-min presentation on the quadratic formula for an intro class , where we are assuming students know both how to factor and how to complete the square. Please comment: Homework Statement O.K. We are given an equation...

Homework Statement How should be the value of a so quadratic equation ax^2-4x+4=0 to have double solutions? A)\;\;2 B)\;\;1 C)\;-1 D)\;-2 Homework Equations The Attempt at a Solution D=b^2-4ac If: D>0\;\;\rightarrow\; {x_1,x_2}\;\rightarrow\;\text{double solutions.}...

Homework Statement L-1(1-(Lambda1+Lambda2)L+Lambda1Lambda2L2)Ft Use quadratic formula to solve for Lambda1 and Lambda2 The Attempt at a Solution I took the equation inside the brackets and solved for L. It turned out that L1=Lambda1 and L2=Lambda2 But when I plugged those...

Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...

Homework Statement Two platforms 50m in height have targets on them. An object succeeds in impacting with both of these. S=ut + ½ at2 can be rearranged into the form: ½ at2 + ut – s =0 This is a quadratic equation. [ ax2 + bx +c =0 ] Use this to calculate: I. The times at which the...

Homework Statement Hi. I've got the matrix from the Quadratic Sieve down to Gaussian Form and I'm wondering how to find the factor base which leads to a square number now. Homework Equations The Factor Base: $${29,782,22678}$$ The original Matrix: \begin{pmatrix} 0 & 0 & 0 & 1\\...