# Tagged Questions

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Solving Cubic when There are Known to be 3 Real Roots

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ ...
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### How to solve the equation $(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$? [closed]

How to solve this equation? $$(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$$
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### Algorithm for complex roots of high degrees.

Is there an algorithm to find complex roots of equations of high degrees? Let's suppose I'm given an even function of degree greater than 6 that does not have real roots, how am I supposed to find its ...
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### Exponential trigonometric function zeroes

How would I find the zero of this exponential function: $f(x)= sin e^x$ After i set the equation to zero, what do i do? Does this involve finding the derivatives?
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### Let $p=a_0+a_1x+…+a_nx^n$ how can I add $a_{n+1}x^{n+1},a_{n+2}x^{n+2},…$ while maintaining some properties?

Let $p_n=a_0+a_1x+...+a_nx^n$ be a polynomial with only real roots. I need a way to algorithmically find real coefficients $a_{n+1},a_{n+2},...\neq 0$ such that the polynomial ...
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### When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
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### “Extending” the calculation of the golden ratio using square roots (not silver-ratio)

I'm looking at the following formula: $x =\frac{-n+\sqrt{n^{2}+4n}}{2}$ For $n=1$ this this gives $0.618...$ and then $\frac n x$ gives $1.618...$ which is $\phi$, the golden ratio. What ...
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If $x^5+ax^4+ bx^3+cx^2+dx+e$ where $a,b,c,d,e \in {\bf R}$ and $2a^2< 5b$ then the polynomial has at least one non-real root. We have $-a = x_1 + \dots + x_5$ and $b = x_1 x_2 + x_1 x_3 + ... 3answers 33 views ### The root of a trigonometrical function [closed] How could I calculate the root of the following function? $$f(x)=2[\arctan(x+3)+\arctan(\frac{x-1}{5})]-\pi$$ 0answers 45 views ### Is it possible to find the zeros of Riemann Zeta function using programming? [closed] I am new at C# language and wondering about this topic. Is it possible to find the zeros? EDIT: The most important point for me; what is the procedure or pseudocode to compute first a few zeros? Does ... 1answer 50 views ### Find number of roots of the equation$e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$Find number of roots of the equation$e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$Using Descartes rule, number of positive roots is zero and there can be a maximum of 4 negative roots. Also, for ... 1answer 57 views ### Find all roots of a polynomial using secant method [closed] I am writing a program that will find all zeros of a polynomial using secant method. But this method despite my best efforts doesn't always converge. How to find the initial value for the secant ... 0answers 29 views ### Solving degree 3 polynomial to get real root Get the real number$z$such that$z^3 - 63z - 162 = 0$using Cardano's method. I haven't learned about Cardano's method how can I get the real number$z$that is a root of the following polynomial? 3answers 44k views ### Quadratic equation - Alpha and Beta Roots If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ... 1answer 432 views ### Inverse Quadratic Interpolation and the secant method I am currently completing a maths project that aims to approximate the roots of functions using MATLAB. The two root finding methods that I have used are inverse quadratic interpolation and the ... 1answer 445 views ### Working with casus irreducibilis I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor$x^3 - 15x - 4$to find$4$as a root and it also has two other real roots. Using Cardano's method we ... 2answers 105 views ### Can the roots of the derivative of the polynomial in complex variable be as close as we want them to be from the roots of the polynomial itself? The (probably) famous Gauss-Lucas theorem states that the roots of the derivative$P'(z)$are contained in the convex hull of the roots of$P(z)$, where$P(z)$is complex variable polynomial. I am ... 1answer 41 views ### Complex root of a real polynomial Let$n\in\mathbb{N}$. We consider, in$\mathbb R[X]$, the polynomial $$C_{n}=X(X+1)^{2n}-2^nX.$$ Determine the values of$n$for which the complex number$i$is a root of$C_n$. My thoughts: ... 2answers 78 views ###$x^3-3x+1=0$has three real roots$x_{1}\;x_{2}\;,x_{3}$. Then what is the value of$\{x_{1}\}+\{x_{2}\}+\{x_{3}\}$? If the equation$x^3-3x+1=0$has three real roots$x_{1}\;x_{2}\;,x_{3}\;,$Where$x_{1}<x_{2}<x_{3}$. Then the value of$\{x_{1}\}+\{x_{2}\}+\{x_{3}\} = \;,$Where$\{x\}$Represent ... 7answers 2k views ### Find all five solutions of the equation$z^5+z^4+z^3+z^2+z+1 = 0z^5+z^4+z^3+z^2+z+1 = 0$I can't figure this out can someone offer any suggestions? Factoring it into$(z+1)(z^4+z^2+1)$didn't do anything but show -1 is one solution. I solved for all roots of ... 2answers 27 views ### How do I find the poles of this difference equation? I have an equation: $$y(n) = 0.634x(n) - 0.634x(n-2) + 0.268y(n-2)$$ I completed a$z$transform and got: $$H(z) = \frac{1-0.268z^{-2}}{0.634 - 0.634z^{-2}}$$ What is the next step to find the ... 2answers 68 views ### My proof that this entire analytic function's range is within epsilon of any complex number, If$f(z)$is an entire analytic function assuming the values$0$and$1$, show that for any complex number$a$and any real number$ϵ>0$there is a point$z_0$such that$|f(z_0)−a|<ϵ$My work: ... 1answer 43 views ### Trying to find splitting fields over Q of$x^{19} -1$I'm trying to find subfields L of C which are splitting fields over Q For$x^{19}-1$I've found the roots, but since you can't express them in exact form I don't see what to do next. 1answer 34 views ### Minimization optimization - where have I gone wrong? Following @littleO's advice, I've set about to minimize$\sum_n ((x-x_n)^2+(y-y_n)^2+(z-z_n)^2-d^2)^2$. Going using an exact Hessian (because the function is smooth definite) as follows:$\textbf{H} ...
Consider for $T\neq k\pi/\omega_i$ and $\omega_1/\omega_2\not\in\mathbb Q$: $$\psi(t,T)=\dfrac{\sin(\omega_1 t)}{\sin(\omega_1 T)}+\dfrac{\sin(\omega_2 t)}{\sin(\omega_2 T)}$$ When $T$ is in the ...