Roots Definition and 978 Threads

Hi , I had to solve a quadratic equation , i got two roots as an answer ( ans= x1 / x2) , and now i need to use one of those answers to complete further tasks like finding y from x+y=c so i need to use x1 and x2 from roots , i was wondering if that's possible and how

Mentor note: Member reminded that some effort must be shown. Real root is 1.858. Just don't know which method to use to find the 4 complex 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 ...

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 $P(Q(x))=0$.

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.

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

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 $x^3+a^3x^2+b^3x+c^3=0$?

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 $x_3$.

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

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...

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

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 $b$.

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...

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?

#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...

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.

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

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

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...

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?

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?

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...

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...

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

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

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...

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...

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

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...

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...

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...

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 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}##...

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?

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 ?

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.

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...

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)

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...

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...

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...

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

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 =...

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...

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...

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

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...

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...

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