Finding roots of a non linear equation

Question

Can the roots of the following equation can be found analytically?

$$f(x)=-x+x^2+5sin(x)$$

I'm of course referring to the non trivial one (besides $x=0$).

Answer 1

This is a transcendental equation and you need a numerical method. Newton method would be the simplest to use.

Graph your function; see where more or less are the roots and stat iterating. It will be quite fast.

Edit

From the plot, we can seee that the root is close to $-\frac \pi 2$. So, to get an approximation, perform a Taylor expansion around this point. This will give $$\frac{1}{4} \left(-20+2 \pi +\pi ^2\right)-(1+\pi ) \left(x+\frac{\pi }{2}\right)+\frac{7}{2} \left(x+\frac{\pi }{2}\right)^2+O\left(\left(x+\frac{\pi }{2}\right)^3\right)$$ which is a quadratic in $\left(x+\frac{\pi }{2}\right)$. Solve it and you get $$x_0=-\frac{1}{56} \left(20 \pi-8+\sqrt{4544-320 \pi -160 \pi ^2}\right)\approx -1.76962$$ while the exact solution is $\approx -1.76968$.

Answer 2

I guess you're asking for a solution using the elementary operations and radicals a finite number of times (otherwise, what's the point in the first place since you would want some closed form eventually). We can expand the $\sin$ term as an infinite series of powers and can manipulate it as we like since it converges nicely. But what you want would mean solving a power series (infinite) in finite radicals. Clearly you can see that this is impossible.

Thus, you will be best served in using some approximation method, many of which will give you a sufficiently fine approximation to the nonzero root.