Root Approximation
\[ \large \color{red}{3}x^{5} \color{blue}{=} \color{red}{1}x^{4} + \color{red}{4}x^{3} + \color{red}{1}x^{2} + \color{red}{5}x + \color{red}{9}\]
The above polynomial has exactly one real root \( \alpha\) . Find \( \lfloor 1000 \times \alpha \rfloor \).
Notation: \( \lfloor x \rfloor \) denotes the greatest integer smaller than or equal to \(x\). This is known as the greatest integer function.
by
Pranshu Gaba
Consider this equation
\[2^x=x^2\]
There are 2 trivial solutions, namely \(x=2\) and \(x=4\)
However, the equation is known to also have a negative solution. Find the negative solution.
Give your answer to 3 decimal places.
by
Julian Poon
Find the value of \(a\) to 3 significant figures, so that \(x\) and \(y\) are distinct real solutions to
\[x+y=100\pi, \\ x^y = y^x = a. \]
by
Soumava Pal
\[ \large x^y = y^x = 43 \]
If \(x\) and \(y\) are distinct positive real numbers satisfying the equation above, find \(\lfloor 100(x+y) \rfloor \).
You are allowed to use computer assistance.
You might refer to Soumava's algorithm.
by
Agnishom Chattopadhyay
\[ f(x) = 9x^8 + 7x^6 + 5x^4 + 3x^2 - 1 = 0 \]
Which option is closest to the largest real root of the equation above?
by
Aloysius Ng