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# Root Approximation

What are the zeroes of a function? When an exact answer isn't attainable, Calculus provides approximation techniques - the same ones that allow calculators to find your roots.

\[ \large \color{red}{3}x^{5} \color{blue}{=} \color{red}{1}x^{4} + \color{red}{4}x^{3} + \color{red}{1}x^{2} + \color{red}{5}x + \color{red}{9}\]

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.

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.

Find the value of \(a\) to \(x^y=y^x=a\), so that the solutions of \(x\) and \(y\) to the equation satisfy,

\[x+y=100\pi\]

Give your answer correct to 4 decimal places or more.

\[ \large x^y = y^x = 43 \]

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.

\[ f(x) = 9x^8 + 7x^6 + 5x^4 + 3x^2 - 1 = 0 \]

Which option is closest to the largest real root of the equation above?

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