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
1
vote
2 answers

Alternate to Vieta's formula to find $a,b,c,d$ in this question.

Question : Roots of the equation $$x^{4}+2 x^{3}-5x^{2}+7x+10=0$$ are $\alpha, \beta, \gamma, \delta$ and that of $x^{4}+a x^{3}+b x^{2}+c x+d=0$ be $\alpha+\beta+\gamma, \alpha+\beta+\delta, \alpha+\gamma+\delta ; \beta+\gamma+\delta,$ then find…
Wolgwang
  • 1,563
1
vote
2 answers

Finding $f$ given that all roots of $x^8 - 4x^6 + 7x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$ are positive real numbers

Assume $a,b,c,d,e,f$ are real numbers such that all the roots of $x^8 - 4x^7 + 7x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$ are positive real numbers. Find all possible values of $f.$ I'm pretty sure that Vieta's plays a vital part in here, but I'm…
questionasker
  • 1,148
1
vote
1 answer

Show there's only one root in an interval

I want to prove that $-3x+7 = \cos x$ only has one root on $\mathbb R$. I've already shown that there is one root in the interval $[2,3]$ using the Intermediate Value Theorem, and I can show that there are no roots wherever $-3x+7 \notin [-1,1]…
jeremy909
  • 1,028
  • 6
  • 24
1
vote
2 answers

Let $x_{n}$ be the positive real root of $(x^{n-1}+2^n)^{n+1} = (x^{n} + 2^{n+1})^{n}$, how to prove that $x_{n} > x_{n + 1}$?

Let $x_{n}$ be the positive real root of equation $$(x^{n-1}+2^n)^{n+1} = (x^{n} + 2^{n+1})^{n}$$ How to prove that $x_{n} > x_{n + 1}$? Actually, $x_{n} > 2$ and I get that $x_{1} = 5, x_{2} \approx 3.5973, x_{3} \approx 3.1033$
Blanco
  • 656
1
vote
2 answers

Is any combination of roots (eg, $\sqrt2+\sqrt3$) a unique number?

Every non negative real number has a unique non negative root, called the principle square root. Does this mean that any combination of roots added will also produce a unique value? For example, $$\sqrt{2}+\sqrt{3} = 1.414213562373095\ldots +…
Perpetual Fool
  • 43
1
vote
1 answer

How to find rational solutions to a set of equations

I have a set of equations to satisfy and I would like to find if they have rational solutions, and if they do, what they are. The equations are: \begin{equation} 1 + \alpha = 3 a \alpha\\ 1 + \beta = 3 b \beta\\ 1 + \gamma = 3 c \gamma\\ 1 + \delta…
Garry
  • 71
1
vote
1 answer

Solution for the roots of $x^4+x^2+1=0$

Is this solution to find the roots of $x^4+x^2+1=0$ correct? $x^4+x^2+1=0$ $x^4+2x^2+1-x^2=0$ $(x^2+1)^2-x^2=0$ $[(x^2+1)-x][(x^2+1)+x]=0$ $(x^2-x+1)(x^2+x+1)=0$ For this equation to be true, either $(x^2-x+1)=0$ or/and $(x^2+x+1)=0$. Using the…
AYA
  • 578
1
vote
3 answers

Real Solution of $4^x + 6^x = 9^x$.

So here is the problem Solve $$4^x + 6^x = 9^x$$ for $x$. I am trying to find its real solution. I was trying in this way!! $$6^x\left( \frac {4^x}{6^x} + 1\right) = 9^x$$ $$\left({\frac 23}\right)^x +1=\left({\frac 32}\right)^x$$ but I'm stuck…
Badshah Khan
  • 376
1
vote
3 answers

Roots of the equation $(x – 1)(x – 2)(x – 3) = 24$

The equation $(x – 1)(x – 2)(x – 3) = 24$ has the real root equal to 'a' and the complex roots 'b' and 'c'. Then find the value of $\frac{bc}{a}$ My approach is as follow $y=f(x)=(x – 1)(x – 2)(x – 3) - 24=0$ $y'=3x^2-12x+11=0$ Solving we get…
Samar Imam Zaidi
  • 8,800
1
vote
2 answers

How to prove an equation with two unknowns is true for some (any) integers

I have seen few math problems online, about solving one equation with two unknowns, (which is not possible as the number of equations should match the number of unknowns), but I thought is there any way to prove that LHS = RHS for some integer…
YouKnowWhoIAm
  • 115
1
vote
2 answers

Two Different solutions for the same equation

$\sqrt{x}\sqrt{x}=4$ then $x=4$ But $\sqrt{x}\sqrt{x}=\sqrt{x^2}$ So, $\sqrt{x^2}=4$ which leads to $|x|=4$. Why is this happening?
user_
  • 253
1
vote
0 answers

How to explain an approximate method of finding square roots?

I would be grateful if someone could explain the following method of calculating a square root. I've numbered all the steps for reference. I found various approximate methods in google. These methods were quite clearly iterating to the nearest…
OldGrantonian
  • 111
1
vote
2 answers

Where is the mistake in the equality of roots?

I have this statement: If $f(x) = \frac{\sqrt[3]{x}}{\sqrt{2}}, g(x) = > \frac{\sqrt[6]{8x^2}}{2}$, with $x < 0$ is $f(x) = g(x)$ ? My attempt was: $(1)$ $f(x) = \frac{\sqrt[3]{x}}{\sqrt{2}} = \frac{\sqrt{2}\sqrt[3]{x}}{2}$ $(2)$ $g(x) =…
ESCM
  • 3,161
1
vote
4 answers

Why the $-2-\sqrt[3]{-7}$ is a complex root???

When I used the Wolframalpha to solve the equation $$x^3+6x^2+12x+1=0$$, the result was as below: My question is "Why the $-2-\sqrt[3]{-7}$ is a complex root???"
user361960
1
vote
2 answers

Find the number of real roots of the equation $54x^4-36x^3+18x^2-6x+1=0$

Find the number of real roots of the equation $54x^4-36x^3+18x^2-6x+1=0$ I entered the equation in desmos.com and no roots were lying below or x=0 lines , hence all roots are imaginary. Using Descartes rule for f(x) 4 sign change occurs , hence 4,2…
Samar Imam Zaidi
  • 8,800
1 2 3
…
20 21