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.

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Roots of biquadratic equation

This question also was a part of my today's maths olympiad paper: If squares of the roots of $x^4 + bx^2 + cx + d = 0$ are $\alpha, \beta, \gamma, \delta$ then prove that: $64\alpha\beta\gamma\delta - [4\Sigma \alpha\beta - (\Sigma \alpha)^2]^2 =…
Vishwesh
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Why this value is not the second root of the equation?

$\sqrt{x+3}=x-2$ Why $\frac{5}{2} - \frac{\sqrt{21}}{2}$ is not root? There is only one restriction: $\sqrt{x+3}$, but $\frac{5}{2} - \frac{\sqrt{21}}{2} > 0$. $x^2-4x+4=x+3$, $x^2-5x+1=0$, $D=25-4=21>0$ $D>0$, =>, $x = \frac{ 5 \pm \sqrt{21} }{…
Dave
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Exitence of a root for a function

I want to prove that $f(\psi)$ defined below, has at least one root. I tried using intermediate value theorem but I could not prove the existence of a root. I would appreciate it if anybody could help. $f(\psi)=\sum_{i=1}^K \frac{\alpha_i…
Shahram Shahsavari
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How to find how many real roots of an equation?

Roots of $ax^2 + bx + c = 0$ are real and positive. $a$, $b$ and $c$ are real. Then $ax^2 + b|x| + c = 0$ has how many real roots? My try: I studied one method where we see how many signs are changing in equation. Then we are able to find real…
user404716
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Principal root of the third root of 8

I feel like I am right on this question but I keep getting it wrong. I have tried inputting 2 and $2+0i$. Am I missing something? Thank you.
MathIsHard
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For a fixed and small $\epsilon$, finding the number of real roots of $x^{2}+e^{-\epsilon x}-2+\sin(\epsilon x)$

I saw the following question in an introduction to applied mathematics exam (this is only the first part of the question): Assume $0<\epsilon\ll1$ . Denote $$ f(x,\epsilon):=x^{2}+e^{-\epsilon x}-2+\sin(\epsilon x) $$ How many real roots does…
Belgi
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Roots of a cubic equation

I have the following equation: $s^3+as+b=0$ Now I want the values for a and b for which the given equation has the following complex roots: $c \pm di$ I don't really care about the remaining root. Any ideas? I have Matlab available, but I don't know…
Pietair
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How to find the number of real roots of the given equation?

The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is (A) $0$, (B) $1$, (C) $2$, (D) infinitely many. Trial: $$\begin{align} 2 \cos \left( \frac{x^2+x}{6} \right)&=2^x+2^{-x} \\ \implies…
A.D
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Proof By Contradiction, Rational Roots

This was an exam question that I got totally wrong and am a bit question. Prove $x^3 + x + 1 = 0$ has no solutions. Prove by contradiction. Assume: $x^3 +x +1 =0$ has at least one rational root. So, what I attempted to do here was solve for $x$.…
Marla
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What is wrong with this false proof? $-1=(-1)^1=(-1)^\frac{2}{2}=((-1)^2)^\frac{1}{2}=\sqrt{(-1)^2}=\sqrt{1}=1$

$-1=(-1)^1=(-1)^\frac{2}{2}=((-1)^2)^\frac{1}{2}=\sqrt{(-1)^2}=\sqrt{1}=1$ This proof bugs me for the following reasons: Mathematicians have defined the symbol $\sqrt{}$ (Named the principal square root) to mean 'take only the positive square root…
Dom Turner
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roots of an equation (infinite)

Is it possible to write a finite equation consisting ONLY of exponentials, logs and $x^n$ where $n$ is any real number which has an infinite number of non-equal real roots ? (trivial examples like $x-x=0$, $(\sqrt(x)^2)$/$x =1$, imaginary numbers,…
Quadratica
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Finding number of real roots of an equation

Equation is--> $$ x^{13} + x - 1/e^x - \sin(x) =0 $$ To find number of real roots of the equation. Context--> I am solving previous years questions of IIT Jam Mathematical Statistics (MS entrance exam) . My approach--> I took $e^{-x}$ and $\sin(x)$…
Rio
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Approximation of square roots

Recently, I've seen a YouTube video where they approximate square roots real quick. They use this approximation : $$\sqrt{x} \approx \lfloor \sqrt{x} \rfloor+\dfrac{x-(\lfloor \sqrt{x} \rfloor)^2}{2\lfloor \sqrt{x} \rfloor}$$ I want to know the math…
ARahman
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Finding how many solutions a function has

I have the function $$f (x) = \cos x - \frac{x^2}{100}$$ $$f'(x) = -\sin x \, -\frac{x}{50}$$ $$f''(x) = -\cos x - \frac{1}{50}$$ and I want to find out how many roots it has, I have tried using calculus and differentiating it. I have tried using…
Guysudai1
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An Elliptic Curve Question

An elliptic curve is an equation of the form $y^2 = x^3 +ax +b$. There are elliptic curves that are one continuous curve, and there are some where there is a loop and a curve. What value of $a$ for a given $b$ gives a loop and a curve that touch…
William Grannis
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