2
votes
0answers
32 views

What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
4
votes
2answers
222 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
3
votes
1answer
42 views

Find zero of sum of 4 modified Bessel functions

I am trying to find the (positive) root of the function $f(x) = I_{-3/4}(x) + I_{3/4}(x) - I_{-1/4}(x) - I_{1/4}(x)$ where $I_\alpha(x)$ denotes the modified Bessel function of the first kind. ...
0
votes
3answers
73 views

How do I solve the trigonometric equation $\sec^3x - 2 \tan^2 x = 2$? [closed]

A friend asked to me how could she resolve this equation, but I don't know how to resolve it?? Could you help me?. The equation is : $\sec^3x - 2 \tan^2 x = 2$ Note: She told me that I can use ...
2
votes
4answers
41 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
7
votes
1answer
85 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
0
votes
3answers
87 views

Using sum/product of quadratic roots to solve cubic equation

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $6x^2 + 2x - 3 = 0$, how do I find the value of: $$ \alpha^3 + \beta^3 $$ and: $$ \frac{1}{\alpha^3} + \frac{1}{\beta^3} $$ ...
2
votes
1answer
55 views

Given roots (real and complex), find the polynomial

This is not a duplicate of theory of equations finding roots from given polynomial. Given that the roots (both real and complex) of a polynomial are $\frac{2}{3}$, $-1$, $3+\sqrt2i$, and $3+\sqrt2i$, ...
1
vote
1answer
54 views

roots of sum of exponential functions

Could anyone point me in the right direction of finding the roots of equations of the form $$ \sum_{i=1}^n a_ie^{f_i(x)}, $$ where $a_i \in \mathbb{R}$ and the $f_i$ are each first degree polynomials ...
0
votes
1answer
51 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
6
votes
2answers
192 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
6
votes
3answers
82 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
6
votes
4answers
128 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
5
votes
1answer
85 views

Find $\lfloor {\alpha}^6 \rfloor$

If $\alpha$ is a real root of the equation $$x^5-x^3+x-2=0$$ find the value of $\lfloor {\alpha}^6 \rfloor$. This one totally stumped me. We are asked to calculate $\lfloor {\alpha}^6 ...
7
votes
3answers
116 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
0
votes
1answer
54 views

Root of equation, solvability

I was trying to solve the following equation for t $$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$ ...
1
vote
1answer
97 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
5
votes
4answers
103 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
4
votes
2answers
64 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
3
votes
2answers
60 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
1
vote
1answer
40 views

Determine all real solutions of the system of n equations

For $n\geq3$, determine all real solutions of the system of $n$ equations: $x_{1}+x_{2}+...+x_{n-1}=\frac{1}{x_{n}}$ ... ...
0
votes
2answers
26 views

Possibility of integral quadratic with these roots

If x and w are the roots of a quadratic equation with integral coefficients then is this possible: ${x = w = \frac{2}{3}}$. The correct answer says it is, but how is that so if it means: ...
4
votes
4answers
191 views

Can $x^3+x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve it: $x^3 + x^2 + 1 = 0$ Am I missing something here, or is indeed a more advanced method necessary to solve ...
3
votes
2answers
57 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
0
votes
4answers
85 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
5
votes
1answer
86 views

A number related to the roots of a quartic polynomial is a root of a cubic polynomial

So here is the problem, $a$ and $b$ are two distinct real roots of $f(x)=0$ where $f(x)=x^4-6x+3$, show that $(a+b)^2$ is a root of $g(x)=x^3-12x-36$. I have tried many methods, such as substitution, ...
1
vote
2answers
60 views

How is the nature of the roots of a third degree polynomial determined?

Given a polynomial $p(x) = x^3-bx^2+cx-d = 0 $ such that all three roots are real positive integers. How does one figure out if the three roots are distinct? The coefficient of $x^3$ is 1. In the case ...
2
votes
5answers
273 views

How to solve $x^4-8x^3+24x^2-32x+16=0$

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$
1
vote
1answer
77 views

Find the solutions of the equation…

How can I solve this equation? $$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$ Frankly, I just have no idea at all!!! Thank you in advance!
1
vote
0answers
154 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
0
votes
2answers
258 views

How to solve for a non-factorable cubic equation?

I want to know how one would go about solving an unfactorable cubic. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for ...
1
vote
2answers
41 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
0
votes
2answers
62 views

Square root of negative integer

Can I write: $-\sqrt{(2)}$ = $\sqrt{(-2)}$ and vice versa? Or, say, we have, $(-\sqrt{(x - 4)}$ Can this be changed into $(\sqrt{(4 - x)}$ by taking the minus sign inside the square root? How?
1
vote
2answers
83 views

How to find the zeros of $f(x) = 2x(5-x)$

How do I get the zeros, if $f(x) = 2x(5-x)$. I have told by my classmate that in order to get the zeros of $f(x) = 2x(5-x)$, I need to distribute $2x$ to $(5-x)$. So I distribute it to make it ...
0
votes
2answers
88 views

Intriguing Equation

How many ordered tuples of 7 integers ${\{x_{i}\}}_{i=1}^{7}$ are there, such that $$\sum _{i=1}^{7}{x_{i}}-\prod_{i=1}^{7}{x_{i}} =6$$ where $1\le x_i\le 8$. I tried taking ${ \{ x_{ i }\} }_{ ...
5
votes
2answers
428 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...
0
votes
6answers
109 views

How to find all solutions of $4^x-3^x=1$?

I have problem with equation: $4^x-3^x=1$. So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other ...
0
votes
3answers
124 views

$P(x)=x^5+ax^4+bx^3+cx^2+dx+e$ has roots $1,2,3,4$ and $k$. Find $P(5) -P(0)$.

A polynomial $P(x)$ with leading coefficient $1$ is of degree $5$, and its distinct roots are $1, 2, 3, 4$ and $k$. Find the value of $P(5) -P(0)$. I have no clue on what my initial steps should be.
2
votes
1answer
95 views

Game of polynomials

Written on a blackboard is the polynomial $x^2+x+2014$.Calvin and Peter take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or ...
2
votes
1answer
108 views

Find the value of $m + n + r$

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$ ,where $m$ is non zero integer and $n$ and $r$ are relatively prime numbers.Then the value of $m+n+r$ ...
0
votes
1answer
69 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
0
votes
4answers
158 views

How can I solve $y^{3}-3y^{2}+2=0$?

I am stuck at this equation $y^{3}-3y^{2}+2=0$. How do I solve it without calculator? It might be very trivial so I think I just need a hint. It is actually a substitution $y=\log x$,but I think it ...
1
vote
1answer
97 views

Rational Root theorem issue

I've given my class an example: $$2x^3+3x^2+6x+4=0$$ By the rational root theorem, if there is a rational root then it should be of the form $\frac{p}{q}$ where $p$ is a factor of 4 and $q$ is a ...
3
votes
3answers
394 views

$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation $\displaystyle ...
0
votes
2answers
166 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
10
votes
4answers
215 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
0
votes
2answers
142 views

A 3rd degree polynomial $P(x)$ has three unequal real roots. What is the least possible # of unequal real roots for $P(x^2)$

I got that if P(x) is a 3rd degree polynomial then P($x^2$) must be a 6th degree polynomial. I don't know how to proceed from here.
2
votes
4answers
128 views

Find the solution of the equation

Find all real solutions of this equation : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
2
votes
2answers
33 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
17
votes
6answers
2k views

Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...