Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [roots]

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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

Questions regarding values $x$, such that a function $f$ evaluates to zero at $x$. For questions about "square roots", "cube roots" and such, consider using the and the tag. For questions about roots of Lie algebras, use the tag instead.

6663 questions
2
votes
1 answer

Help on solving the equation $\frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{a+x}}=\frac{\sqrt{a-x}}{\sqrt{a}-\sqrt{a-x}}$

Could you give me some help on finding the roots (if any) of the following equation: $$ \frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{a+x}}=\frac{\sqrt{a-x}}{\sqrt{a}-\sqrt{a-x}} $$ I tried to apply some classic approaches, but I had no luck... Could you lend me…
nullgeppetto
  • 3,006
2
votes
1 answer

Unique solution of algebraic equation

Prove that $10^x+11^x+12^x=13^x+14^x$ has an unique solution over $\mathbb R$. By inspection the equation is true for $x=2$
Achari S Ganesha
  • 898
2
votes
2 answers

Roots of a sixth degree polynomial

I have this question: The polynomial $f(x) = x^6 - ax^4 - ax^2 +1 $ has $(x-p)$ as a factor, where $a,p$ are real numbers. Show that $a = p^2 + p^{-2} - 1$ Here's my attempt: Let $u = x^2 \implies f(x) = u^3 - au^2 - au + 1$ Let $\alpha, \beta,…
imulsion
  • 869
2
votes
3 answers

Prove $x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT.

Prove $f(x)=x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT. What I did: $f(-1)=-3$ $f(0)=1$ As $f$ is continuous, there exists a $c \in (-1,0) /f(c)=0$ Then computed $f'(x)=7x^6+15x^4$. But $f'\ngtr0 \forall…
YoTengoUnLCD
  • 13,384
2
votes
4 answers

Find the square root of $404.11$.

Find the square root of $404.11$ without using calculator accurate upto $2$ decimal places . It is clear that $20<\sqrt{404.11}<21$ so it will be $20.ab$ without trial and error what could be the fast way to compute it .
R K
  • 2,635
2
votes
1 answer

If $z$ is an $n$th root of unity, prove that $1/z$ is an $n$th root of unity

I'm not sure if how I'm going to prove this to be correct: Since $z$ is an $n$th root of unity, it means $z^n = 1$ For $1/z$ to be an nth root of unity, lets take it to the power of $n$, $(1/z)^n$, and so $1/(z^n) = 1/1 = 1$, hence, $1/z$ is an…
Joe
  • 91
2
votes
1 answer

Write a biquadratic equation that has as roots the numbers $2$ and $2\sqrt{2}$

I thought the answer would be: $$(x^2 - 4)(x^2 - 8) = 0$$ but it has $4$ roots the positive and negative values. Which is the correct answer?
prishila
  • 23
2
votes
4 answers

Given that the equation, $(k-1)x^2-2(k-1)x-(3k+1)=0$ has real roots, show that $k^2-k≥0$

I can get to $k^2-k≥0$ but only when I make $b^2$ negative. The problem is why would I make $b^2$ negative other than the fact that $b$ is negative in the original equation? The problem with this is that $c$ is also negative and so I would also have…
Thomas Winkworth
  • 641
2
votes
2 answers

Find coefficient in quartic given product of roots

The product of two of the roots of $$x^4 -11x^3 + kx^2 + 269x - 2001=0$$ is $-69$. Find k. This is a question I have recently received, and I am required to take a test on related questions tomorrow. I have absolutely no idea on how to start with…
Shreyash Chaudhari
  • 147
2
votes
1 answer

For any positive number $k$, find the value of $1^k + 2^k + 3^k+...+(p-1)^k$(mod $p$)

For any positive number $k$, find the value of $1^k + 2^k + 3^k+...+(p-1)^k$(mod $p$) and prove that your answer is correct. A Little confused about this problem. Any help? Would love to see a solution for this. Thank you.
Pasie15
  • 491
2
votes
1 answer

root of sum of exponentials

I am curious to know which values of $t \gt 0$ solve the following equation in terms of the constants $a,b,c$. $a e^{-2 b t} - e^{-2 t} + c e^{-3 b t} + c e ^{- 3 t} = 0$ where $a \gt 1, b \gt 1, c \gt 0, t \gt 0$. I would like to know how many…
monomnomnomial
  • 23
1
vote
1 answer

problem regarding theory of equations

given quadratic equation : ${x^2+bx+c=0}$ let the roots of the equation be ${u}$ and $v$. let ${S_0 = u^0+v^0}$ let ${S_1 = u^1+v^1}$ let ${S_2 = u^2+v^2}$ show that : ${S_2+bS_1+S_0 = 0}$
samarth srivastava
  • 125
1
vote
2 answers

Sum of fifth power of roots

What is the sum of fifth power of roots of the equation $$x^3+qx+r$$.I tried expanding $$(a+b+c)^5$$ but it didn't work instead it is becoming more and more complex.
nat
  • 11
1
vote
1 answer

primitive roots, field dimension

Let $\zeta$ be a primitive $m$-th root of $1$. Determine the values of $m$ such that: $[\mathbb Q$($\zeta$):$\mathbb Q$]$=2$. The only thing I have in mind is that $[\mathbb Q $($\zeta$):$\mathbb Q$]=$\varphi (n)$= Euler $\varphi$ fuction. But I am…
Buzi
  • 407
1
vote
1 answer

Solution of equations of the form: $a^x+b^x+c=0$

Is it possible to solve equations of the form: $a^x+b^x+c=0,\;abc\neq0$ with analytical methods; if so, how is this done?
Natanael
  • 157
1 2 3
…
20 21