What is Roots: Definition and 976 Discussions

The Roots are an American hip hop band, formed in 1987 by Tariq "Black Thought" Trotter and Ahmir "Questlove" Thompson in Philadelphia, Pennsylvania, United States. The Roots serve as the house band on NBC's The Tonight Show Starring Jimmy Fallon, having served in the same role on Late Night with Jimmy Fallon from 2009 to 2014.
The Roots are known for a jazzy and eclectic approach to hip-hop featuring live musical instruments and the group's work has consistently been met with critical acclaim. ThoughtCo ranked the band #7 on its list of the 25 Best Hip-Hop Groups of All-Time, calling them "Hip-hop's first legitimate band."In addition to the band's music, several members of the Roots are involved in side projects, including record production, acting, and regularly serving as guests on other musicians’ albums and live shows.

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  1. chwala

    Find the cubic equation given the roots

    ##\sum ∝=3## ##\sum ∝β=0## ##∝βγ=-4## ##\sum2 ∝=6## ##\sum 2∝.2β##=4##\sum ∝β=0## ##2∝.2β.2γ=-32## we then end up with ##x^3-6x^2+0x+32=0## ##x^3-6x^2+32=0## i am looking for alternative methods ...
  2. anemone

    MHB Real Roots of Composite Polynomials: Solving P(Q(x))=0

    Let $P(x)=x^2+\dfrac{x}{2}+b$ and $Q(x)=x^2+cx+d$ be two polynomials with real coefficients such that $P(x)Q(x)=Q(P(x))$ for all real $x$. Find all real roots of $P(Q(x))=0$.
  3. fresh_42

    Constructive Proofs (open) Boundaries on the roots of splitting real polynomials

    Suppose all roots of the polynomial ##x^n+a_{n−1}x^{n−1}+\cdots+a_0## are real. Then the roots are contained in the interval with the endpoints $$ -\dfrac{a_{n-1}}{n} \pm \dfrac{n-1}{n}\sqrt{a_{n-1}^2-\dfrac{2n}{n-1}a_{n-2}}\,. $$ Hint: Use the inequality of Cauchy-Schwarz.
  4. anemone

    MHB Finding $k$ for 2 Non-Negative Roots of $x^2-2x\lfloor x \rfloor +x-k=0$

    Find all values of $k$ for which the equation $x^2-2x\lfloor x \rfloor +x-k=0$ has two distinct non-negative roots.
  5. anemone

    MHB Real Roots of Cubic Equation: $x^3+a^3x^2+b^3x+c^3=0$

    An equation $x^3+ax^2+bx+c=0$ has three (but not necessarily distinct) real roots $t,\,u,\,v$. For what values of $a,\,b,\,c$ are the numbers $t^3,\,u^3,\,v^3$ roots of an equation $x^3+a^3x^2+b^3x+c^3=0$?
  6. anemone

    MHB Roots of Cubic Equation: Finding $x_1,\,x_2$, and $x_3$

    The roots $x_1,\,x_2$ and $x_3$ of the equation $x^3+ax+a=0$ where $a$ is a non-zero real number, satisfy $\dfrac{x_1^2}{x_2}+\dfrac{x_2^2}{x_3}+\dfrac{x_3^2}{x_1}=-8$. Find $x_1,\,x_2$ and $x_3$.
  7. anemone

    MHB Can the polynomial equation $x^8-x^7+x^2-x+15=0$ have real roots?

    Prove that the polynomial equation $x^8-x^7+x^2-x+15=0$ has no real solution.
  8. chwala

    Solving a quadratic equation as a sum and product of its roots

    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}##...
  9. S

    Value of "a" between two real roots of quadratic equation

    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...
  10. anemone

    MHB Finding $b(a+c)$ for Real Roots of $\sqrt{2014}x^3-4029x^2+2=0$

    Let $a>b>c$ be the real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $b(a+c)$.
  11. anemone

    MHB Max Value of $b$ for Real Roots of $f(x)$ and $g(x)$

    Let $a$ and $b$ be real numbers and $r,\,s$ and $t$ be the roots of $f(x)=x^3+ax^2+bx-1$ and $g(x)=x^3+mx^2+nx+p$ has roots $r^2,\,s^2$ and $t^2$. If $g(-1)=-5$, find the maximum possible value of $b$.
  12. G

    MHB Quadratic equations intersaction point is minimum instead of roots

    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...
  13. G

    MHB Are Critical Points and Roots Interchangeable in Math?

    Might be I am asking a silly question but really want to clarify that would critical points and roots are same terms use interchangeably? I mean we can use critical point as value of x and root also as value of x then what is the difference between?
  14. A

    Comp Sci Recursive Double code to Calculate the sum of the square roots <= a number

    #include<stdio.h> #include<math.h> double foo(int n){ if(n==1){ return(1); } if(n!=0){ return( sqrt((n)+foo(n-1) ) ); } } int main(){ int num; printf("Enter the number: "); scanf("%d",&num); foo(num); printf(" %lf ",foo(num)); return(0); }I...
  15. anemone

    MHB Real Roots of Polynomial Minimization Problem

    For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.
  16. anemone

    MHB Roots of a Polynomial Function A²+B²+18C>0

    If a polynomial $P(x)=x^3+Ax^2+Bx+C$ has three real roots at least two of which are distinct, prove that $A^2+B^2+18C>0$.
  17. D

    Calculate the roots of -t^2-2t+1

    I have to compute the roots in order to compute an integral partial fraction decomposition ##\frac {2 \pm 2 \sqrt {4+4}} {-2} = -1 \mp \sqrt 2## the correct on is ## (t+1- \sqrt 2)(+1+ \sqrt 2) \\or \\ (t+1+ \sqrt 2)(+1- \sqrt 2)## the general rule is
  18. C

    I Question about the roots of Chebyshev polynomials

    Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as tk=cos(πk/N) k=0, ..., N...
  19. Akash47

    Finding roots of an exponential equation

    I know that both 3^x and e^x can't be 0 for real x. Then x^2-4=0 is the only choice and we get x=2,-2. Am I right? Or should I add something?
  20. K

    Proving f'(x) Properties & Finding Its Roots: A Challenge

    1.Prove that f'(x) is strictly decreasing at (- ##\infty##,a) and strictly increasing at (a,##\infty##). 2.Prove that f'(x) has exactly two roots. I tried to find f''(x)=0, but I'm not able to solve the equation. What should I do?
  21. AutGuy98

    MHB Find n such that the group of the n-th roots of unity has exactly 6 generators

    Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...
  22. AutGuy98

    MHB Prove that the 12-th roots of unity in C form a cyclic group

    Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...
  23. P

    MHB Find the cubic equation that has -1 and 2i as roots

    Answer is given, but no explanation or logic for it. From HiSet free practice test
  24. Waffle24

    Quadratic equation -- question about the roots

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

    B Properties of roots of polynomials

    i have some doubts from chapter 1 of the book Mathematical methods for physics and engineering. i have attached 2 screenshots to exactly point my doubts. in the first screenshot...could you tell me why exactly the 3 values of f(x) are equal. the first is the very definition of polynomials...but...
  26. R

    B Roots of Polynomials: Understanding Mathematical Methods

    I was reading this book - " mathematical methods for physics and engineering" in it in chapter 1 its says "F(x) = A(x - α1)(x - α2) · · · (x - αr)," this makes sense to me but then it also said We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met if (1.8) were...
  27. binbagsss

    I Quartic real roots - factor part into quadratic

    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
  28. G

    I What method should I use to get the roots of this equation?

    Mentor note: Thread moved to Diff. Equations subforum Hello, few days ago I had a calculus test in which I had to find the general solution for the next differential equation: (D^8 - 2D^4 + D)y = 0. "D" stands for the differential "Dy/Dx". However I could only find 2 of the roots on the...
  29. Monoxdifly

    MHB [ASK] Exponents and Roots Simplification problem

    The result of \frac{7x-\frac92\sqrt[6]{y^5}}{\left(x^{\frac56}-6y^{-\frac13}\right)x^{-2}} for x = 4 and y = 27 is ... a. \left(1+2\sqrt2\right)9\sqrt2 b. \left(1+2\sqrt2\right)9\sqrt3 c. \left(1+2\sqrt2\right)18\sqrt3 d. \left(1+2\sqrt2\right)27\sqrt2 e. \left(1+2\sqrt2\right)27\sqrt3 I got...
  30. M

    MHB Gaussian Quadrature: isolated roots

    In an exercise I have determined the Gaussian Quadrature formula and I have applied that also for a specific function. Then there is the following question: Explain why isolated roots are allowed in the weight function. What exacly is meant by that? Could you explain that to me? What are...
  31. M

    I Do complex roots have a physical representation on a curve?

    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...
  32. Adgorn

    B Questions regarding polynomial divisions and their roots

    Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...
  33. WMDhamnekar

    MHB Euler equations having double roots as a solution

    If the Euler equations have double roots as it's solution, second solution will be $y_2(x)=x^r\ln{x}$. what is its proof? or how it can be derived?
  34. TachyonLord

    I Where does the exponential function come from in roots?

    For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
  35. A

    MHB Find the roots of f(x)+g(x)+h(x) = 0.

    Let f(x) , g(x) and h(x) be the quadratic polynomials having positive leading coefficients an real and distinct roots. If each pair of them has a common root , then find the roots of f(x)+g(x)+h(x) = 0.
  36. S

    American & Canadian PF members only: colonial roots?

    As a follow-up to a previous thread I started about % of white Americans with colonial roots, I thought I'd pose the following question below, about whether any of you (American or Canadian PF members), as far as you know, have roots in what is now Canada or the US dating back to the 17th or...
  37. E

    MHB Find the square roots of 4*sqrt(3)+4(i)

    So I have a study guide for my final which was written by a different professor from my actual professor. So I don't understand the question, I don't know if it's because my professor did not teach this or if the wording is different from what I'm used to: Find the square roots of 4*sqrt(3)+4(i)
  38. J

    Using binomial coefficients to find sum of roots

    Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
  39. J

    MHB Using Rolle's theorem to prove for roots (part 2)

    Hi, I have done up the proof for the question below. Please correct me if I have done wrong for the proof. Thanks in advanced!Question: Prove that if ab < 0 then the equation ax^3 + bx + c = 0 has at most three real roots.Proof: Let f(x) = ax^3 + bx + c. Assume that f(x) has 4 distinct...
  40. J

    MHB Using Rolle's theorem to prove for roots

    I have deduce a proof as stated below and am not sure if it is correct, therefore need some advice. Question: Prove that if ab > 0 then the equation ax^3 + bx + c = 0 has exactly one root by rolle's theoremProof: Let f(x) = ax^3+bx+c = 0. f(x) is continuous and differentiable since it is a...
  41. Hafsaton

    Prove: Sq Root of a Sum ≤ Sum of the Sq Roots

    Homework Statement (x.y)ER+ that means x and y >=0 Homework Equations Prove that n√(x+y)<=n√x + n√y The Attempt at a Solution
  42. J

    Partial Fraction Expansion - Repeated Roots Case

    Homework Statement Find Partial Fraction Expansion 10/[s (s+2)(s+3)^2] Homework EquationsThe Attempt at a Solution 10/[s (s+2)(s+3)^2] = A/s + B/(s+2) + C/(s+3)^2 + D/(s+3) A = 10/[(s+2)(s+3)^2], s approaches 0 = 10/(2*3^2) = 5/9 B = 10/[s (s+3)^2], s approaches -2 = 10/(-2) = -5 C =...
  43. alejandromeira

    B Exploring Olbers' Paradox in Matt Roots' Introduction to Cosmology

    I'm beginning to study the Matt Roots book Introduction to Cosmology and in the section 1.3 Olbers' Paradox he writes: "If the surface area of an average star is A, then its brightness is B=L/A. The sun may be taken to be such an average star, mainly because we know it so well. The number of...
  44. Mr Davis 97

    Prove that the roots of unity is a cyclic group

    Homework Statement Let ##\mu=\{z\in \mathbb{C} \setminus \{0\} \mid z^n = 1 \text{ for some integer }n \geq 1\}##. Show that ##\mu = \langle z \rangle## for some ##z \in \mu##. Homework EquationsThe Attempt at a Solution My thought would be just to write out all of the elements of ##\mu## in...
  45. V

    MHB Laguerre polynomials Roots

    Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. LαN(xi)=0
  46. Subrahmanyan

    To find the nature of roots of a quintic equation....

    The asks for us to find the nature of roots of the following equation ,i.e,rational or irrational nature of the roots: the Equation is : http://www5a.wolframalpha.com/Calculate/MSP/MSP7106108efdacf0gb84ie000039f29d33b6hie7ic?MSPStoreType=image/gif&s=56&w=63.&h=18.x^5+x=5 I have been able to...
  47. T

    I Proof that cube roots of 2 and 3 are irrational

    Proof by contradiction that cube root of 2 is irrational: Assume cube root of 2 is equal to a/b where a, b are integers of an improper fraction in its lowest terns. So the can be even/odd, odd/even or odd/odd. The only one that can make mathematical sense is even/odd. That is...
  48. M

    MHB How to Understand the Ratio of Quadratic Roots?

    86 https://uploads.tapatalk-cdn.com/20180712/7faa109f841501a10a2d07b6e8e10681.jpg
  49. M

    MHB Sum and product of roots of quadratic equation

    #85
  50. opus

    B Confusion about The Conjugate Roots Theorem

    As a preface to this theorem stated in my text, it states that: "If all the coefficients of a polynomial ##P(x)## are real, then ##P## is a function that transforms real numbers into other real numbers, and consequently, ##P## can be graphed in the Cartesian Coordinate Plane." It then goes on...
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