Quadratic Definition and 990 Threads

  1. F

    MHB Solve Quadratic Equation: 2E(S+Wn)2

    so I've been tasked with this insane question obtain the solution of the following quadratic equation how do I solve this, the 'E' (greek letter),is throwing me out,my complete hazardous guess is 2E(S+Wn)2 <--- 2 is a squared sorry my keyboard skills are lacking also 'tut' any help...
  2. T

    B Quadratic term's relationships

    I know everyone here must know this but it is a new revelation to me and very interesting. But I have a question about the reason behind the following... Take ##x^2+10x+10## for example. Any coefficient of ##x^2## modifies the "zoom" of the parabola, for lack of a better word, and determines...
  3. E

    MHB Expression containing coefficients of a quadratic equation

    Let $a$ be the smallest root of the equation $x^2-9x+10=0$. Find $a^4-549a$. Extra credit if the solution does not find the actual roots of the equation.
  4. KevinMWHM

    Al-Khwarizmi's 6th quadratic case

    The problem: I need to come up with a formula based on Al-Khwarizmi's 6th algebraic equation; bx+c=x^2. I'm just having a definition problem that's holding me up from moving forward. The first line of his solution is to "halve the number of roots". What is meant by "number of roots"? Number of...
  5. mr.tea

    Quadratic forms under constraints

    Homework Statement Find the minimum value of ## x_1^2+x_2^2+x_3^2## subject to the constraint: ## q(x_1,x_2,x_3)=7x_1^2+3x_2^2+7x_3^2+2x_1x_2+4x_2x_3=1 ## Homework EquationsThe Attempt at a Solution I am not really sure how to think about it. I have seen the opposite way but have not seen this...
  6. G

    Solutions for a quadratic system of equations

    Hi, I'm looking for the real solutions to the system \begin{array}{rcl} 1 & = & v_1+v_2+v_3+v_4+v_5 \\ 1 & = & v_1^2+v_2^2+v_3^2+v_4^2+v_5^2 \end{array} Background: I'm looking at a Newton's cradle with 5 balls, each of mass ##m=1##. The first ball is pulled away and let go such that it hits...
  7. M

    Quadratic and linear Intersection

    Homework Statement How many points of intersection does the line y = x have with the parabola y = 6x − x2 −10 ? A) none B) one C) two D) three E) more than three Homework EquationsThe Attempt at a Solution x = 6x - x^2 - 10 -x^2+5x - 10 Used b^2 - 4ac to check the number of intersections. The...
  8. B

    Can a 3x3 Matrix Represent a Quadratic, Cubic, or Quartic Function?

    I have a doubt... Look this matrix equation: \begin{bmatrix} A\\ B \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix} = \begin{bmatrix}...
  9. emrys

    Solving Quadratic Equations - (x2+3x+3)1/3 + (2x2+3x+2)1/3 = 6x2+12x+8

    Homework Statement (x2+3x+3)1/3 + (2x2+3x+2)1/3 = 6x2+12x+8 2.Relevant equationsThe Attempt at a Solution (x2+3x+3)1/3 -1 +(2x2+3x+2)1/3 -1 = 6x2+12x+6 (x2+3x+2)/((((x2+3x+3)1/3)2 + (x2+3x+3)1/3 +1) + (2x2 +3x+1)/((((2x2+3x+2)1/3)2)+(2x2+3x+2)1/3 +1) -6(x+1)2=0 then x=-1 or...
  10. S

    MHB How to fit a quadratic function to two points?

    Hi! I am trying to build a better understanding of quadratic functions. So I had this idea of trying to fit a quadratic function to two arbitrary points, but I am not sure how this really works. Given the points (-4,1) and (12,1) on a Cartesian plane, how can a quadratic function be defined...
  11. J

    Why can linear approximation equal quadratic approximation

    Hi I'm having trouble visualizing why in a function such as 1/(1-x2) linear approximation of 1/(1-u) where u = x2 is the same as quadratic approximation of 1/(1-x2) The linear approximation is 1+u or 1+x2 Quadratic approximation is the same, 1+x2 Can someone explain to me why this happens...
  12. R

    MHB Quadratic Equation Roots and Coefficients: Solving for Unknowns

    $\alpha$ and $\beta$ are the roots of the equation $2{x}^{2}-5x+c=0$. If $4\alpha-2\beta=7$, find the value of $c$. I did the following: $\alpha+\beta=-\frac{-5}{2}=\frac{5}{2}$ $\alpha\beta=\frac{c}{2}$ $\frac{c}{2}=\frac{7+2\beta}{4}\cdot\frac{-7+4\alpha}{2}$...
  13. D

    Finding the quadratic equation that suits the solutions

    Homework Statement if x1 and x2 are solutions to 1) x² + x + 1 = 0, then 2) y1 = ax1 + x2 and 3) y2 = x1 + ax2 are solutions to which quadratic equation? Homework Equations ax² + bx + c = 0 x1∕2 = (−b ± √(b² - 4ac))/2a The Attempt at a Solution Well, firstly i solved for x1 and x2 getting...
  14. R

    Find D: [(A - D)^P] / [(B - D)^R] = (C - D)^(P - R)

    How do I get D by itself? This one's got me baffled [(A - D)^P] / [(B - D)^R] = (C - D)^(P - R)
  15. S

    Invariance of quadratic form for unitary matrices

    Homework Statement Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##. Homework Equations The Attempt at a Solution ##|x'|^{2} = (x')^{\dagger}(x')...
  16. S

    Invariance of quadratic form for orthogonal matrices

    Homework Statement Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##. Homework Equations The Attempt at a Solution ##x'^{2} = (x')^{T}(x') =...
  17. M

    Rationalize denominator & factorising quadratic equations.

    Homework Statement Homework Equations Not Sure. The Attempt at a Solution For the first question I know you have to multiply the conjugate of the denominator so it would be (2 - √5)/(1−2√5) x (1−2√5)/(1−2√5) but I'm not sure how to actually do that. For the second question. I have that...
  18. astrololo

    Calculating the roots of a quadratic with complex coefficien

    Homework Statement Calculating the roots of a quadratic with complex coefficients Homework Equations x^2 - (5i+14)x+2(5i+12)=0 The Attempt at a Solution I tried the quadratic solution but it gives too complicated solutions. I have no idea on how to do this...
  19. L

    Understanding Quadratic Inequality: Explained in Detail

    Can someone explain to be in detail what is quadratic inequality? It's rather confusing. Thank you
  20. moondaaay

    Solve Quadratic Inequality: x²-4x+3≤(3x+5)(2x-3)

    Homework Statement How to solve this kind of inequality? x²-4x+3≤(3x+5)(2x-3)Homework EquationsThe Attempt at a Solution :[/B] I'm confused. Should I factor the left side or should I FOIL the right side then equate it to zero to find the critical numbers? Help pleaasee.
  21. B

    Is There a Fundamental Theorem of Calculus for Squared Derivatives?

    Fundamental theorem of calculus: \int_a^b \frac{df}{dx} dx = f(b) - f(a) All right? Everybody knows... BUT, BUT, exist some analog of the FTC for this case: \int_a^b \left( \frac{df}{dx} \right)^2 dx = ? Hum?
  22. J

    Quadratic drag — a baseball is thrown upwards

    Homework Statement [/B] A baseball of mass m is thrown straight up with an initial velocity v0. Assuming that the air drag on the ball varies quadratically with speed (f = cv^2), show that the speed varies with height according to the equations. Where x_{0} is the highest point and k = c/m...
  23. icystrike

    Quadratic Inequality: Solving for x | No Quotes

    Homework Statement [/B] As attached Homework EquationsThe Attempt at a Solution [/B] The answer is stated as option A. However, my solution is -6≤x≤3; I can seems to find an option that fits the solution.
  24. M

    MHB Can Absolute Values of Quadratic Functions Determine Their Discriminants?

    Let f(x) and g(x) be quadratic functions such as the inequality \left| f(x) \right| \ge \left| g(x) \right| is hold for all real x . Prove that \left| \Delta_f \right| \ge \left| \Delta_g \right|. For quadratic function f(x)=ax^2+bx+c , then \Delta=b^2-4ac. I have no idea how I could...
  25. E

    Quadratic forms and kinetic energy

    I heard that proportionality of kinetic energy with square of velocity, ##E_k\propto v^2##, can be derived with help of quadratic forms. It goes like: we guess that ##E_k\propto v^2## and we assume that momentum ##p\propto v##, then equation is valid in another inertial system. And so on. The...
  26. karush

    MHB What is the quadratic trig identity for cosine when simplified?

    $$\cos\left({4x}\right) =8\sin^4\left({x}\right) -8\sin^2\left({x}\right) +1$$ I thought this would break down nice from the quadratic but it didn't.
  27. B

    Analyzing the coefficients of the quadratic equation

    Is possible classify the quadric equation Axx + Bxy + Cyx + Dyy + Ex + Fy + G = 0 how straight, hyperbola, circle, ellipse, parabola, etc, in the same way that is did in the phase plan: https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg...
  28. T

    MHB Can someone give a simple explanation of quadratic residues?

    Hi, This is not coursework, just private study. Ok, I understand that q is a quadratic residue MOD n if x^2 = q MOD n What I don't understand is how to figure this out? I read a paper that states "8 is a quadratic residue mod 17, since 5^2 = 8 MOD 17", fair enough. It then goes on to state...
  29. D

    Factoring a quadratic: strange issues

    Homework Statement Hello! One of the easiest rules (when possible to apply) to factor a quadratic is to find both x-s by x1 + x2 = b x1 * x2 = c Homework Equations Please, take a look at what is written in the book. I can't grasp why x1 = -2 and x2 = 3, and not, as I thought, x1 = -3 and x2...
  30. K

    MHB Proving Prime Numbers in Quadratic Imaginary Fields

    Hi, I need your help with the next two problems: 1) If p is a prime number such that p\equiv{3}\;mod\;4, prove that \sqrt{-p} is prime in \mathbb{Z}[\sqrt[ ]{-p}] and in \mathbb{Z}[\displaystyle\frac{1+\sqrt[ ]{-p}}{2}] too. 2) 2) We have d > 1 a square-free integer. Consider the quadratic...
  31. diracdelta

    Quadratic form and diagonalization

    Homework Statement Find diagonal shape of next quadratic form ( using eigenvalues and eigenvectors) Q(x,y)= 5x2 + 2y2 + 4xy. What is curve { (x,y)∈ ℝ| Q(x,y)= λ1λ2, where λ1 and λ2 are eigenvalues of simetric matrix joined to quadratic form Q. Draw given curve in plane. The Attempt at a...
  32. Keen94

    The discriminant of a quadratic and real solutions

    Homework Statement Given a series of mathematical statements, some of which are true and some of which are false. Prove or Disprove: 1. A sufficient condition that ax2+bx+c=0 (a≠0) have a real root is that b2-4ac>5. 2.A necessary condition that ax2+bx+c=0 (a≠0) have a real root is that...
  33. manogyana25

    Quadratic equations - problems

    A train travels a distance of 480km at a uniform speed. If the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train. Represent this situation in the form of quadratic equation. So I have been trying this but couldn't find...
  34. binbagsss

    Tricky Quadratic formula / trig identities

    Im to solve ##(k+l)^{2}e^{-ila}-(k-l)^{2}e^{ila}=0##, for ## k##, The solution is ##k=l(e^{ial}-1)/(e^{ial}+1)=il tan(al/2)## FIRST QUESTION So it's a quadratic in k, should be simple enough, my working so far using the quad. formula is ##k= (4l^{2}(e^{-ila}+e^{ila})\pm...
  35. S

    If a and b are both quadratic residues/nonresidues mod p & q

    Homework Statement If a and b are both quadratic residues/nonresidues mod p & q where p and q are distinct odd primes and a and b are not divisible by p or q, Then x2 = ab (mod pq) Homework Equations Legendre symbols: (a/p) = (b/p) and (a/q) = (b/q) quadratic residue means x2 = a (mod p) The...
  36. E

    Is xTAx always non-zero for a real, symmetric, nonsingular matrix A?

    Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true. True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.
  37. C

    Given quadratic equation how are the integer solutions found?

    Not homework but given the question it probably fits here best Given the following equation $$x^2+138x+317=y^2$$ How do you find the integer solutions? For example wolframalpha has the solutions. but I cannot see how they are derived http://www.wolframalpha.com/input/?i=x^2+138x+317=y^2 By...
  38. Y

    Modeling Quadratic Air Resistance in 2-D using Mathematica

    Homework Statement I am working on a little project in which I analyze a video I found online and try to determine if it is real of a hoax. The video I am analyzing includes a man throwing a football off of a football stadium (at the very top) and making it into a basketball hoop at field...
  39. C

    MHB Finding a Quadratic Function from a Vertex

    Hello, I'm having some trouble on this question, and I'd imagine it's just because I'm looking at it incorrectly. The problem statement is: The quadratic function f(x) = p + qx - x2 has a maximum value of 5 when x = 3 (i.e. vertex at (3, 5), right?) Find the value of p and the value of q...
  40. D

    Difficulty solving the following quadratic equation

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > Hi, I'm having difficulty solving the following quadratic equation x^2-(660/x-5)-91=0. The fact that the middle term has an x-5 rather than just an x that is throwing me Can anyone please help...
  41. moriheru

    Quadratic Residues: Explanation & Modulo Odd Prime Number

    This may be a bit vague but can anyone explain this sentence to me http://en.wikipedia.org/wiki/Quadratic_residue:"Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues." If this is to vague I apologize.
  42. M

    Calculating velocity of a bullet with quadratic air drag

    So this problem was a 2 part question. The first part goes as such. 1. A gun is fired straight up. Assuming that the air drag on the bullet varies quadratically with speed, show that the speed varies with height according to the equations: v2 = Ae-2kx - (g/k) (upward motion) v2 = (g/k) -...
  43. Valerie Prowse

    Relating quadratic graphs & bouncing movement

    Homework Statement Hi everyone, I am doing a lab currently (and it looks like a few other people on here have had similar questions...) and I'm having trouble with one of the concepts. I had to drop a ball from a height and measure the movement with a motion sensor, which was graphed with the...
  44. M

    MHB Help with Quadratic Equations by completing the square

    I understand how to solve these equations when the square is on this side of the equal sign: x2 + 8x + 7 = 27 But when the square is on the other side, I am thrown. Like this one... x2 = 14x - 33 The solutions manual shows the next step as the following, but what do you do to get to this...
  45. L

    MHB Therefore, the roots of the given equation are $x=-4$ and $x=2$.

    Solve (x+3)^2 = 4x+17 where did i go wrong? (x+3)(x+3 )= 4x+17 x^2 + 3x + 3x + 9 = 4x+17 x^2 + 6x + 9 = 4x + 17 x^2 + 6x + 9 - 4x - 17 = 0 x^2 + 2x - 8 = 0 (x-2)(x+4) <-- USING THE CROSS METHOD x= -2, 4 is my cross method working incorrect or something? do explain my mistake
  46. N

    Solving a Quadratic Equation: Finding the Minimum

    Homework Statement The Attempt at a Solution Looks like the graph would be a parabola? And since it's a>0 it would be upward and therefore a minimum. Not sure what that would be. Unsure how to solve the rest[/B]
  47. LiHJ

    Quadratic Inequality: Solve for 4/m-2/n Without a Calculator

    Homework Statement Dear Mentors and PF helpers, Here's the question: The roots of a quadratic equation $$3x^2-\sqrt{24}x-2=0$$ are m and n where m > n . Without using a calculator, a) show that $$1/m+1/n=-\sqrt{6}$$ b) find the value of 4/m - 2/n in the form $$\sqrt{a}-\sqrt{b}$$ Homework...
  48. T

    Algorithm to find square root of a quadratic residue mod p

    I'm going through an explanation in a number theory book about Tonelli's algorithm to find the square roots of a quadratic residue modulo ##p## where ##p## is prime, i.e. I want to solve ##x^2 \equiv a \pmod{p}## with ##(\frac{a}{p}) = 1##. The book goes as follows: Let ##p - 1 = 2^s t##, where...
  49. N

    Standard form, vertex form. Something isn't right here

    I mainly just need some clarification here. I was doing my homework and then browsing the web to find an answer to my problem and came across mathewarehouse' definition of Standard form and then I looked at my homework and went..."huh?" I don't understand if my homework is listening this wrong...
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