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
28
votes
6 answers

Function with no roots

Given a non-constant function $f(x)$, is it possible for it to have no zeroes (neither real nor complex)? Say for example, $f(x)=\cos x-2$, does a complex solution exist for this because for real $x$, $\cos x$ belongs to $[-1, 1]$? Or, say…
Aditi Narware
  • 490
7
votes
1 answer

Numerical inverse of logarithmic integral

What is the best way to numerically calculate the inverse of the logarithmic integral, defined by $$ \operatorname{li}(x)=\int_{0}^{x}\dfrac{1}{\log(t)}\operatorname{d}t $$ eg $\operatorname{li}(x)=100,\ x=?$
martin
  • 8,998
6
votes
2 answers

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$?

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$? I tried this way: Let $$f(x)=\sqrt{4+\sqrt{4-x}}$$ So, $x=f^2(x)=f^{2n}(x)$ where $n\in\mathbb{N}$. Then, I tried to prove that $f^k(x)=f^{k+1}(x)$ for any…
user164524
6
votes
2 answers

Is there a simpler method of calculating $\sqrt[n]{x}$?

I've began to reteach myself Algebra and am brushing up on my roots (pun intended). I've been following this website (which makes it simple to refresh your memory), and as I reviewed the root examples I couldn't help but wonder if there was a simple…
Hazel へいぜる
  • 459
6
votes
3 answers

Why four roots to this equation: $(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$

$$(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$$ I figured the roots are $0$, $1$, $-1$, and $9$. But why?
chx
  • 1,807
5
votes
2 answers

How to solve for the roots of an equation of two variables ? For example $\frac{4L^6-19L^4x}{9}+\frac{8L^2x^2}{3}-x^3$.

I want to solve for the roots of an equation which has two variables in it. For example: $$24L^2x^2+ 4L^2x-9x = 0$$ This is just an example which I made up. Mine is a bit more complicated. I am not used dealing with this sort of thing, so is there…
this guy123
  • 129
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  • 9
5
votes
2 answers

Why does this method for finding cube roots work?

While on the Internet I came across a formula for cube root using recursion. The formula was: $$ use \space x_1 = \frac{a}{3} \space (a \space is \space the \space number \space we \space want \space to \space find \space cube \space root \space…
Suhrid Mulay
  • 217
5
votes
2 answers

Finding roots of equation

$$r\left(\frac 1p -1\right)- \left(\frac 1p\right)^\frac89 + \left(\frac 1p\right)^\frac19 =0$$ where $ r = 2^\frac89 - 2^\frac19$ How do I solve this without a computer?
damson_jam
  • 65
5
votes
5 answers

Real Roots and Differentiation

Prove that the equation $x^5 − 1102x^4 − 2015x = 0$ has at least three real roots. So do I sub in values of negative and positive values of $x$ to show that there are at least three real roots? The method to do this question is not by finding the…
user271716
  • 131
5
votes
1 answer

Proof of why $\sqrt[x]{x}$ is greatest when $x=e$

Stated above question. If the mathjax I used was wrong, it should be: Why does the xth root of x reach the greatest y at x=e
AAron
  • 413
5
votes
3 answers

condition for a cubic to have a repeated root

To write a condition for a cubic to have 2 real roots, can I equate the fuction to its derivative? I.e let $y=ax^3+bx^2+cx+d$ $\frac{dy}{dx}=3ax^2+2bx+c$ setting y and the derivative equal to 0 gives: $ax^3+bx^2+cx+d=0$ $3ax^2+2bx+c=0 \Rightarrow…
user221707
  • 61
5
votes
3 answers

$x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$.

I need to prove that the equation $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$. How would I go about proving this?
GWAO
  • 67
4
votes
4 answers

Showing a function has one root in an interval

Could anyone shine some light on this question please? By considering $f'(x)$, show that $$f(x)=x^3 - 2$$ has exactly one root for $x$ greater than or equal to $0$.
gary
  • 41
4
votes
4 answers

Find roots of $3^x+x^3=17$

Find $x$ in the following equation: $$3^x+x^3=17$$
voca
  • 141
4
votes
1 answer

About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Let $z$ be a complex number. Consider $f_n(z)=\sum_{k=1}^n k^{-z}$. Now I wonder : Are there infinitely many positive integer $n$ such that there exists a $z$ with $f_n(z)=0$ and $Re(z)>1$ ? I know that such $z$ exists for $n=13$.
mick
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