Quadratic Definition and 990 Threads

I used the identity ##\sqrt{x^2}=|x|## and completed the square as follows: \begin{align*} x^2-3|x|-2&=0 \tag1\\ \sqrt{x^4}-3\sqrt{x^2}-2&=0 \tag2\\ (\sqrt{x^2}-\frac{3}{2})^2-\frac{9}{4}-2&=0 \tag3\\ (\sqrt{x^2}-\frac{3}{2})^2&=\frac{17}{4} \tag4\\...

I am quite unsure where to post this, as I am a chemical engineer and that I am working with phase equilibrium. My issue however is regarding statistical mechanics, thus me posting my problem here. I keep on encountering that the second virial coefficient of a mixture must be quadratically...

Hi all, I've been trying to follow a question I came across on a website. And I'm able to understand everything up until the separation of variables for solving the differential equation and coming to a solution with arctan. But there are a few things that aren't explained that I was hoping...

But the answers to these two questions confuses me. For the first question the answer goes like: $$m\ddot y = -mkv^2-mg$$ $$\therefore \ddot y = v\frac{dv}{dy}$$ $$\therefore v\frac{dv}{dy}= -kv^2-g$$ but for the answer to the second question we have: $$m\dot v = mg - cv^2.$$ Both questions...

Hi, I'm trying to solve this integral and then isolate V, but I can't get the right answer. I don't know where is my errors. I probably muffed the integral. ##-bv -cv² = m\frac {dv}{dt}## ## \int_0^t dt = - m \int_{Vo}^v \frac {dv}{bv+cv^2} ## I get this after the integration ##t =...

Given are two equations: $$S_1 = ax^2+2hxy+by^2 + c=0$$ $$S_2 = a'x^2+2h'xy+b'y^2 + c'=0$$ This source states that there are several methods to solve for ##x## and ##y##. One of them is the following quote:"Treat equation ##S_1## as a quadratic equation in ##x## and solve it for ##x## in terms...

The graph's turning point of a quadratic function f(x)=ax^2+bx+c is over the X-axis. If the coordinate of the turning point is (p, q) and a > 0, the correct statement is ... A. c is less than zero B. c is more than zero C. q is less than zero D. q equals zero Since the point (p, q) is over the...

my original working, i would appreciate alternative methods...

for the sum, ##\frac {1}{∝^3}##+##\frac {1}{β^3}##=##\frac {β^3+∝^3}{∝^3β^3}## =##\frac {(∝+β)[(∝+β)^2-3∝β]}{∝^3β^3}## =##\frac {-b}{a}##...

For quadratic equation to have two real roots: b2 - 4ac > 0 (-2 (2a + 1))2 - 4 (2) (a (a - 1)) > 0 4 (4a2 + 4a + 1) - 8a2 + 8a > 0 16a2 + 16a + 4 - 8a2 + 8a > 0 8a2 + 24 a + 4 > 0 2a2 + 6a + 1 > 0 Using quadratic formula, I get a < (-3 - √7) / 2 or a > (-3 + √7) / 2Then how to know if a...

y=A/[B+(x-C)^2]

Suppose the system of equations (coming from invariance of the wave equation) : $$B=-vE\\A^2-B^2/c^2=1\\E^2-c^2D^2=1\\AD=EB/c^2\\B=vA\\AE-BD=1$$ If one adds a lightspeed movement like $$A=a+f\\B=b-cf\\D=d+h\\E=e-ch$$ Then solving equ 1 for f gives $$f=(b+ve-vch)/c$$ Equ 4 for h implies...

I derive the quadratic form of Dirac equation as follows $$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\psi=0$$ And I need to find the form of the spin dependent term to get the final expression $$g...

I have 2 quadratic functions and I am interested in their root in the specific range. I use quadratic equation to get their roots and what I find that if their any real solution exist for both or any of the function that lie in it designated specific range, then the roots are maximum or minimum...

Slightly confused at what it wants me to do here?

A recent https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-411-apr-5th-2020-a-27196.html#post119308 asked about properties of a pair of positive integers $x$, $y$ such that $2x^2+x = 3y^2+y$. But it is not obvious that any such pairs exist. So the challenge...

I am doing an investigation of how much a beam sags, based on the distance from its midpoint. This is my hypothetical equation: The relationship between distance, d and sag, s is not a linear relationship. Below, is the determined relationship between the variables, linearized by natural...

I chose coordinates where down is positive. So the force going up is $$F_{up} = mg - cv^2$$ $$a = g + \frac{c}{m}v^2$$ $$a = g + \frac{c}{m}v^2$$ $$a = g \left(1 + \frac{v^2}{v_t^2}\right)$$ $$a = \frac{dv}{dt} = v\frac{dv}{dy} = g \left(1 + \frac{v^2}{v_t^2}\right)$$ I used normal separation of...

##z^2\leq x^2+y^2, z\geq x^2+y^2## I know the shapes of those inequalities, but the question is: How do i find if the point are external the shape or internal?

I was thinking of this simple equation here, ## x^2 = 4##. Many students present the solution as follows. $$ x^2 = 4 $$ $$ \therefore x = \sqrt{4} = \pm 2 $$ Now, even though the final answer is correct, there is a mistake in arriving at the solution. Square root symbol means that we have to...

By working with the following definition of minimum of a quadratic form ##r(\textbf{x})##, ##\lambda_1=\underset{||\textbf{x}||=1}{\text{min}} \ r(\textbf{x})## where ##\lambda_1## denotes the smallest eigenvalue of ##r##, how would one tackle the above problem? It is clear that the diagonal...

I just ran this pretty quick at work. But this is the general outline. Sorry for the slop, it will get better with time. Thanks in advance and any additional info can be supplied. The material is from the Khan free course. ..... A student is fed up with doing her kinematic formula homework...

Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...

Hi, Can anyone help me understand how I get to the answer on this one? The diagram shows a sketch of the graph of y = x2 + ax + b The graph crosses the x-axis at (2, 0) and (4, 0). Work out the value of b.Thank you in advance!

Summary: could you explain why this equality is a quadratic form identity? i read this equality (4.26) here w depends on two variables. it is written that if B is bounded (L2) then it is a quadratic form identity on S. what does it mean? is it related to the two variables? next the author...

## \frac{-3x^4}{7}+x=\frac{-3x^4}{7}+10 \\x=10## I solved the equation and got x=10, My question is how can I solve for x without brute-force method( Multiply all terms by (x-13)(x-7)(x-14)(x-6) and solving for x)

I am having so much trouble figuring this out, I would really appreciate some help. The question is: The following function, L, gives the approximate percent literacy rate in India t years after 1900. L(t)=5.3 x 1.025^t Which of the following equivalent functions shows, as a constant or...

Problem Statement: ##2x^2 + y^2 = 4## Max and min of ##4x + y^2## ? Relevant Equations: F'(x) = 0 To find critical point X = 0 -> y = +- 2 Y =0 -> x = +- ##\sqrt 2## I input that Find f(xy) = 4 and f(xy) = +-##4\sqrt 2## To find with derivative i find both x and y are zero The answer is wrong

Maybe there's a formula or rule that I'm not aware of...:rolleyes:

If I have ##f(x)=x^4+(x+2)(x+1)## basically a quartic without a cubic term for which it can be written as above : ##x^3## + some quadratic which has discrimant ##\geq 0 ##, so that the quadratic has real roots, can one ocnclude that ##f(x)## has real roots too? thanks

Hi PF! Given the quadratic eigenvalue problem ##Q(\lambda) \equiv (\lambda^2 M + \lambda D + K)\vec x = \vec 0## where ##K,D,M## are ##n\times n## matrices, ##\vec x## a ##1\times n## vector, the eigenvalues ##\lambda## must solve ##\det Q(\lambda)=0##. When computing this, I employ a...

Sample Problem 2.04 Drag race of car and motorcycle I was following all the way up to using the quadratic equation for this problem...(please see img for a more detailed attempt at a solution.) So I may have simplified incorrectly here: but I came up with 2.8t^2-(58.8)t+408.1=0 but when...

Quadratic equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0 is (a) elipse when ##B^2-4AC<0## (b) parabola when ##B^2-4AC=0## (c) hyperbola when ##B^2-4AC>0## I found this in Thomas Calculus. However for some values of parameters ##A=17##, ##C=8##, ##B=\sqrt{4 \cdot 17 \cdot 8}##, ##D=E=0##, ##F=20## I got just...

Hi! I am aware that standard fitting numerical methods like Levenberg-Marquardt, Gauss-Newton, among others, are able to fit a dataset z = f(x,y) to a quadratic surface of the form z = Ax2 + Bxy + Cy2 + Dx + Ey + F, where A to F are the coefficients. Is there a simpler method that exists? I'm...

So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much. 1. The quadratic equation whose roots are k less than the roots of ax^2+bx+c=0 is a(x+k)^2+b(x+k)+c=0. 2. The quadratic equation whose roots are k more than the...

−2x^2+3x+20why is this equation so special it is in standard form, and when i solve for zeros i factor down to -1( 2x^2-3x-20) factor ( 2x^2-3x-20) factor by grouping 2(-20)=-40 what numbers multiple to equal -40 and add to equal -3 -8(5) = -40 -8+5= -3 and then just to find zeros the...

If we have y=x^2 -4. This is represented by curve intersect x-axis at (-2, 0) and (2, 0) or if we wish to find it algebraically we set y =0 then we solve it. The roots must lie on the curve. when y=x^2+4 the roots are 2i and -2i "complex" consequently there is no intersection with x-axis, so...

Hello, In general, any equation is a statement of equality between two expressions. In the 2D case, equations generally involve the two variables ##x## and ##y## or either variable alone if we require the other variable to be equal to zero. The most general quadratic equation should be the...

5. Which of these quadratic functions has exactly one x -intercept? o A. y=x 2 −9 o B. y=x 2 −6x+9 o C. y=x 2 −5x+6 o D. y=x 2 +x−6 A 2. What are the x-intercepts of y=(x−2)(x+5) ? o A. (0, 2) and (0, -5) o B. (0, -2) and (0, 5) o C. (-2, 0) and (5, 0) o D. (2, 0) and (-5, 0) D 5. Which of...

Homework Statement Let ##a,b## be squarefree integers and set ##R = \mathbb{Z}[\sqrt{a}]## and ##S = \mathbb{Z}[\sqrt{b}]##. Prove that a) There is an isomorphism of abelian groups ##(R,+) \cong (S,+)##. b) There is an isomorphism of rings ##R\cong S## if and only if ##a=b##. Homework...

Homework Statement ##x,y,z \in \mathbb{R}##, find the minimal value for $$x^2+2y^2+z^2-6x+4y-10z+17$$ Homework Equations None The Attempt at a Solution First I try to use”complete the square” method to make the polynomial something like: $$(x-3)^2+(\sqrt2y+\sqrt2)^2+(z-5)^2-19$$ Then I am...

Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0 ##...

Hey this Quadratic application question is giving me trouble. A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200km/h. If the difference in the times of the flights was 2 hours, what was the jets speed from Bangkok to Tokyo? Just need the...

Homework Statement If ##x## is a rational function of ##y##, such that (ax2 + bx + c)y + (a'x2 bx' + c') = 0 prove that (ac' - a'c)2 = (ab' - a'b)(bc' -b'c) Homework Equations The quadratic formula The Attempt at a Solution This equation can be rewritten as: (ay + a')x2 (by + b')x (cy +c') = 0...

Justify the following by using table, graph and equation. use words to explain each representation f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)

Here's the problem I was given: Kyle is driving at a constant 45.0 m/s when he passes his street racer friend, Cameron. After a 4.00 second delay to get the car started and into gear, Cameron starts chasing Kyle with a constant acceleration of 2.00 m/s/s. How far will Cameron have to drive to...

Homework Statement For which values of b will the quadratic function ƒ(x) = x2-2bx+7 have a minimum value of 6? Homework Equations y = ax2+bx+c y = a(x-h)2+k b(first one) = -2ah(second one) c(first one) = ah2+k(second one) The Attempt at a Solution...

Homework Statement Find the slope of ##y=x^2+4## at (-2,8) and the equation for this line. Homework EquationsThe Attempt at a Solution This problem is intended to give an intuition on how limits work and I think I get the general idea. If we want to find the rate of change (or slope) of some...

I can't come up with this function for hours: I just started to learn quadratic functions so there must be an obvious solution for this which I clearly missed...