What math do I need to solve this: 0.5x^2 - 2x + lnx = -1.687?

[tmoney120]

TL;DR Summary

How do you solve an equation that has both a quadtratic and logarithmic term?

I am looking for information on this type of equation but I don't know the terminology so I could not find anything for these types of equations:

0.5x^2 -2x + lnx = -1.687

What math do I need to solve something like this algebraically?

 

[member 587159]

Such things are not really solvable algebraically. I.e. you won't be able to get an answer in terms of standard functions.

You might be able to express in answer in terms of the Lambert-W function though. See https://en.wikipedia.org/wiki/Lambert_W_function

 

[mfb]

You won't get a nice algebraic solution. Even for a much simpler case, x+ln(x)=c there is no such thing (with exceptions, of course). There are solutions with functions that basically say "f(x) is the solution. Here is your solution: f(x)" and there might be solutions that involve some limit processes.

 

[tmoney120]

Thanks! I thought so, but wanted to see if I missed something obvious, maybe complex numbers or something, but anyway, thanks and see you around the forums!

The answer is 0.351 by the way, and I got that by graphing the function.

 

[Mark44]

The answer is 0.351 by the way, and I got that by graphing the function.

No, that's an approximation to the answer. The other people responding to your question said that the problem isn't solvable algebraically (except possibly by the Lambert W function), but one can use numerical means, including looking at a graph, to get a reasonably close approximation.
Wolframalpha gives the solution as .351307, which is again an approximation, and not the exact solution.