# Questions tagged [algebraic-equations]

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### $3(2x+d)+c(x+5)=10x+17$

Given "$3(2x+d)+c(x+5)=10x+17$" what are the values of c and d. Upon expansion we get $6x+3d+cx+5c=10x+17$, meaning c must be equal to 4. I was playing around with the equation and I found ...
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### Inference of unknown Vandermonde-Matrix

Given some $\{y_m\}_{m=1}^\infty$ and $n$ what is the minimum $d>0$ in order to deduce $x \in \mathbb{R}^n$, where $x$ solves: $$y_m = \sum_{j=1}^n x_j^m \quad \forall m \in 1\dots d$$ or in ...
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I have been thinking about this equation: $$x^2=2^x$$ I know there is two integer solutions: $x=2$ and $x=4$. But there also is a negative solution, that is approximately $x=-0.77$. $$(-0.77)^2=0.5929... 1 vote 2 answers 51 views ### Proving 2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)=3(a^2+b^2+c^2+(a+b+c)^2)^2 in another way? How do I prove the following identity without expanding both sides directly.$$2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)\\=3(a^2+b^2+c^2+(a+b+c)^2)^2$$I expanded both sides directly and it ... -4 votes 1 answer 78 views ### I love maths but I am lazy to read maths undergraduate textbooks in the least [closed] I started developing passion for mathematics back in 2013 when my Visual Basic Programming language tutor asked me and two other students to write a program to compute the factorial of any positive ... 2 votes 1 answer 38 views ### Need help simplifying a set of equations (and understanding how to solve it) i have three algebraic expressions, each using the others. in these equations a, b, c and <... 4 votes 4 answers 315 views ### Roots of a certain sixth order polynomial I am looking for the roots (or basically any information regarding them) of the sixth order polynomial$$p(x):=ax^6+(a+1)x^4+2bx^3-b^2$$for positive, real constants a,b. Since p(0)=-b^2<0 and ... 1 vote 3 answers 190 views ### How to solve Quadratic Equations with an Unknown C and other variables. For instance 3x^2 - 11x + r, I understand the value of r is 6 through trial and error but trial and error is extremely inefficient and time consuming thus not useful in exam situations, How ... 11 votes 2 answers 416 views ### What's wrong with manipulating this algebraic equation? and why does a manipulated system of equations have a different solution than the original? I'll give an example for my first question: x^2 + x + 1 = 0 Clearly x = 0 and x = 1 aren't solutions, so first we can safely divide by x: x + 1 + 1/x = 0 By subtracting 1/x from both sides ... 5 votes 2 answers 179 views ### Solve 2x^2+y^2-z=2\sqrt{4x+8y-z}-19 I am trying to solve the following equation.$$ 2x^2+y^2-z=2\sqrt{4x+8y-z}-19 $$To get rid of the square root, I tried squaring both sides which lead to$$ (2x^2+y^2-z+19)^2=16x+32y-4z $$which was ... 0 votes 1 answer 40 views ### Simplify a.b+c.d Suppose that, (S, +, \cdot) is a semiring, where the operations are defined as x\cdot y=min(x, y) and x + y=max(x, y). Can we further simply the expression a\cdot b+c\cdot d?, where a, b, c, ... 0 votes 1 answer 26 views ### Resolve 1 equation with 3 unknowns with specific condition for each unknown I want to find the possible solution of the following problem: T_k= 21600 seconds 1281.49 < T < 21600 (T in seconds) x and y don't have unit x and y have to be whole number x > 0... 2 votes 1 answer 58 views ### If \sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05, how can we find k? Let n be any natural number, let k\in\{0, \dots, n\}, and let p \in [0, 1]. If \sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05, how can we find k (in terms of n and p)? 0 votes 0 answers 33 views ### Expanding algebraic equations of polynomials. So suppose we have G(z+h,w+t) where G is an algebraic equation such that G(z,w)=g_0(z)+g_1(z)w+\cdots+g_m(z)w^m, each g_i(z) is a polynomial of z \in \mathbb{C}, and w is any function. My ... -3 votes 1 answer 44 views ### Solve 3^x+x^3=17. [duplicate] It's a question I find a bit difficult to solve, and the question is:$$3^x+x^3=17.$$1 vote 0 answers 33 views ### How can a field L be a subgroup of the Galois group of the extension L:K? I am studying the notion of a Galois group, and a remark in my notes is the following: "L:K an extension. Let H be a subgroup of Aut(L). Let L^{H} be the fixed field of H. If L \leq G(L:... 1 vote 1 answer 222 views ### Show that, if a, b, c are real numbers and ac = 2(b + d), then at least one of x^2 + ax + b = 0 and x^2 + cx + d = 0 has real roots. Show that, if a, b, c are real numbers and ac = 2(b + d), then, at least one of the equations x^2 + ax + b = 0 and x^2 + cx + d = 0 has real roots. I've have tried many times and used ... 0 votes 1 answer 78 views ### How to find the roots to the polynomial z^3+z^2+iz^2-z+1+3i I'm having a bit of a problem on how to compute forward the values for z from this equation:$$ z^3+z^2+iz^2-z+1+3i=0 $$In the question there's given that one solution or root is z=i, by solving ... 0 votes 1 answer 50 views ### vertical displacement I have the below HW question: What I've done is replace t with 2, thus obtaining that the ball is at a height of 1.25m after 2 seconds and hence, it will travel over the net. Am I correct? 0 votes 0 answers 50 views ### On multiple roots in algebraic closure of a given field I’m having trouble digesting a fact that popped up in a proof from Lang’s Algebra, page 247, proposition 6.1. Given an element \alpha algebraic over k, and its minimum polynomial f(X) over k, ... 0 votes 1 answer 18 views ### Problem Following Algebra Steps I am trying to follow along an answer to a particular question and don't understand one of the steps taken. I don't understand how they got from step 3 to step 4? The steps can be seen below. I ... 0 votes 2 answers 38 views ### Problem Following Along Algebra Steps I have been given a question with the solution. However I am having trouble following the maths steps that have been taken in the solution. I'm unsure how they have gotten from step 1 to step 2, ... 3 votes 4 answers 107 views ### Is there a better way to solve this equation? I came across this equation: x + \dfrac{3x}{\sqrt{x^2 - 9}} = \dfrac{35}{4} Wolfram Alpha found 2 roots: x=5 and x=\dfrac{15}{4}, which "coincidentally" add up to \dfrac{35}{4}. So I'm ... 0 votes 2 answers 143 views ### Why the definition of Galois group in rotman’s Advanced modern algebra not considering repeated roots The definition is provided in the upper figure. What if f(x) had repeated roots? Are the roots considered deduplicated such that there’ are less than n roots f(x) have? As @reuns pointed out, I have ... 0 votes 2 answers 67 views ### Algebraic solution for a system of algebraic equations? I was asked about solving algebraically the following system of algebraic equations.$$f(a,b):=a(1-b)+ab\frac a{a+b}.u = f(a,b),\quad v = f(b,a).$$Solve algebraically (a,b) in terms of (u,v) ... -1 votes 1 answer 54 views ### The real numbers a,b,c,d satisfy a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}. Find the value of d. [duplicate] The real numbers a,b,c,d satisfy a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}. Find the value of d. I've been given this question for a class I'm taking and I'm not really sure where to start. I let \sqrt{a+... 0 votes 1 answer 38 views ### Solve the equation for x. solve the following equation for x (x^2-8x-3)÷(8x-3)=(x^2+4x+4)÷(4x+4) The problem here I am encountering with is figuring out which method to be used here for solving. I am not getting how to ... 0 votes 3 answers 56 views ### Solve (1+x)^2=A\sqrt{1+Cx} for x. x>0, A>0 and C>1. I am trying to come up with a closed form expression for x, even if it is an approximation. Any help appreciated. 3 votes 1 answer 77 views ### how to handle a "Stiff" algebraic equation numerically? I have a question of great practical importance for me, but I would like to ask it on a bit more of a theoretical mode, because I feel I lack the basic knowledge on it. I would also like to mention ... 4 votes 1 answer 148 views ### Solving 8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2} Solve the equation$$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$I have this idea: set$$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$So$$x-1=b^2 , 2a^2+6b^2 =8b-4$$and$$x^2+x-2 =a^2b^2$$and ... 1 vote 0 answers 50 views ### If x is algebraic over a quotient field K of A, then there exists an integral element cx for some A \ni c \neq 0. Let A be a commutative ring, K its quotient field and x algebraic over K. This means that there exists a polynomial f(X) with coefficients in K such that f(x) = 0. In other words, if ... 0 votes 1 answer 86 views ### upper and lower limits for finding just one (real) solution of an algebraic equation (degree 5 and lower) While programming an equation solver (using trial and error), I came across the fact that there are multiple real and complex bounds in which all of the solutions should be. For my case, only real ... 2 votes 1 answer 51 views ### Solution to Mx-x^2=0 where x^2 is the square of the elements of vector x I have been trying to find the solutions for$$Mx=x\circ x$$where \circ is the element wise product. One solution is x=0. But there is another solution x\neq 0, if M has a real positive ... 0 votes 3 answers 82 views ### Solving algebraic equations involving factorials - without trial and error. The question is defined as follows:$$\frac{(x!)^3}{x}-1=3455$$I first did the basics which was getting rid of the 1 onto the left and getting rid of the x, then I factorised both sides to get an ... 4 votes 1 answer 356 views ### Applying the quadratic Tschirnhausen transformation As per my previous question, I attempted to take dxiv's approach, though I can't seem to make much headway. Considering the simpler problem x^3=x+a and the substitution y=x^2+mx+n, I got the ... 1 vote 1 answer 68 views ### Proving non-existence of real roots Prove that for any real numbers a_{85}, a_{84}, a_{83},\dots ,a_3 the equation a_{85}x^{85}+ a_{84}x^{84}+ a_{83}x^{83}+\dots +a_3x^3+3x^2+2x+1=0 has no real roots. (This problem is stated in ... 0 votes 4 answers 212 views ### Solving a symmetric equation involving three variables a,b and c Solve :$$\frac{x+a^2}{a+b}+\frac{x+b^2}{b+c}+\frac{x+c^2}{c+a}=2(a+b+c)$$I am trying to find a simple technique to solve this equation as there is a pattern in the equation, but I could not do. Any ... 0 votes 1 answer 71 views ### approximating irrational roots of algebraic equations with the Pierce expansion Let  p (x) = 1 - x \lfloor \frac{1}{x} \rfloor  then the Pierce expansion of a real number x \in {R} is expressed by \begin{equation} x_1 = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \prod_{m = 1}^n ... 0 votes 2 answers 137 views ### Solving x^2 y''(x) + 6x y'(x) - 9y(x) =0 with similar techniques that are used to solve algebraic equations Consider the ODE$$x^2 y''(x) + 6x y'(x) - 9y(x) =0 .$$It is clear that we can solve the ODE by the method of reduction of order. However, if we "see" the function y as some constant just for a ... 1 vote 0 answers 22 views ### Does this equation imply separation of variables? I have the following equation: \begin{equation} \dfrac{q(t,r)}{s(t,r)}=B(r), \qquad \qquad (1) \end{equation} with q>0 and s>0. Apart from the trivial case q(t,r)=\mathrm{const.} \times ... 0 votes 1 answer 26 views ### Equations of Value Unknown interest rate i have two question in some assessments, both based on "Equations of Value Unknown interest rate" and two irregular contributions. I've been over the textbook, and looking along time on line found ... 0 votes 3 answers 104 views ### Is -1 sum of squares in the field \mathbb{Q(\beta)} where \beta = 2^{1/3}e^{2\pi i /3} Prove that −1 is not a sum of squares in the field \mathbb{Q(\beta)} where \beta = 2^{1/3}e^{2\pi i /3} My attempt : In fact \Bbb Q(\beta) and \Bbb Q(2^{1/3}) are naturally isomorphic. So I ... -3 votes 2 answers 51 views ### Two problems on solving equations in rational and integers [closed] prove that x^2-2y^2=7 has infinitely many solutions in integers. a and b are two non-zero rational numbers.Then if the equation ax^2+by^2=0 has a nonzero rational solution (x_0,y_0)... 4 votes 1 answer 1k views ### Explanation of the Tschirnhausen transformation I am studying the resolution of the quintic equations, which involves the so-called Tschirnhausen transform. The idea is to cancel the fourth and third degree coefficients by a change of variable of ... 2 votes 2 answers 302 views ### How to find all nonnegative integers x, y, z and w such that 2^x3^y-5^z7^w=1 Find all nonnegative integers x, y, z and w such that 2^x3^y-5^z7^w=1. I think they are (x,y,z,w)=(1,0,0,0),(1,1,1,0),(3,0,0,1),(2,2,1,1), but I couldn't prove its sufficiency (or there may ... 1 vote 1 answer 250 views ### Find positive integer solutions of cubic equation with three variables Find positive integer solutions of equation$$t^3 -at^2 + bt - c=0,$$where$$a^3-6ab+7c=0$$(a, b, c are positive integers too). I've tried to use find solutions in modular arithmetic, but there ... 1 vote 2 answers 56 views ### Show that \frac {1}{\sqrt{5}}[(\frac {1}{x+r_+}) - (\frac {1}{x+r_-}) = \frac {1}{\sqrt{5}x}[(\frac {1}{1-r_{+}x}) - (\frac {1}{1-r_{-}x})]  I need to manipulate this equation:$$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) $$to show that$$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) = \...
Let $G$ be an abelian group, and let $n:G\times G\rightarrow \mathbb{Z}$ be a map satisfying for all $u,v,w\in G$ $$n(u+v,w)+n(u,v)=n(u,v+w)+ n(v,w)$$ and $n(0,x)=n(y,0)=0$ for all $x,y\in G$ Is $n$...