# Can all equations with one unknown x be re-arranged to x = …

Apologies for being rusty..

I was recently looking at the calculation of yield to maturity for bonds and was wondering why people in the industry resort to a trial error approach when calculating them.

The YTM calculations boil down to equations like these, where i is the YTM (Source: https://www.investopedia.com/terms/y/yieldtomaturity.asp):

So my specific question is

1) Can the above equation be re-arranged to a form i = ... where I can calculate the right side numerically?

and my general question would be

2) Can't all equations with a single unknown be transformed into i = ... ?

• Consider the equation $42=x\cdot e^x$ with $x$ as the unknown variable. You cannot solve this equation by using only elementay functions. – mrtaurho Aug 30 '18 at 16:50
• This equation, moreover, probably amounts to a quintic polynomial in $i$, which has been proven not to be solvable (in general). – Adrian Keister Aug 30 '18 at 16:51

You can note that $(1+i)$ is to the power 5. In the complex world, there are 5 solutions to the equation $(1+x)^5 = a$. If you are dealing with positive reals (assuming i is a positive reals), there is only one solution (sometimes none). So we will assume that $i$ is a positive or zero real.