Numerical Solution of Equations

Find the roots of the equation $x^{3} - 7x - 7 = 0$ to four decimal digits.

Find the roots of the equation $x^{3} - 8x + 15 = 0$ to four decimal digits.

Find the roots of the equation $x^{4} - 3x^{2} + 4x - 1 = 0$ to four decimal digits.

Find the roots of the equation $x^{6} - 3x^{2} + x - 1 = 0$ to four decimal digits.

Compute to within $10^{-5}$ the root of the equation $x\ln x = 100$.

Compute to within $10^{-5}$ the root of the equation $x = \cos x$.

Compute to within $10^{-5}$ the root of the equation $x^{2} = 20\sin x$.

Compute to within $10^{-5}$ the root of the equation $3x = e^{x}$.

Find the smallest positive root of the equation $x = \operatorname {tg}x$.

Find the smallest positive root of the equation $\operatorname {ch}x\cos x = 1$.

Find the smallest positive root of the equation $1 - \dfrac {x^{2}}{1!^{2}} + \dfrac {x^{4}}{(2!)^{2}} - \dfrac {x^{6}}{(3!)^{2}} + \ldots = 0$.

By one of Ramanujan's identities,

Hence the number $e^{-2\pi }$ is a root of the equation

where $a = \dfrac {1}{24} - \dfrac {1}{8\pi }$. Using this, find the equality: $e^{-2\pi } = x = a - 3a^{2} + 14a^{3} - 82a^{4} + 546a^{5} + \ldots$