how to find $x$ in the following formula?

Question

For a software project, I need to calculate some things. One of the formulas looks like$$26280 = 2 \sqrt{\frac{149598000000 - x}{1.4}} + \frac{x}{10217588910387300000}.$$

My colegue says you can't solve the above equation (you won't be able to find $x$). I am quite convinced that you should be able to find $x$; the problem is, I don't know how :(

I came as far as rewriting the above into$$\left ( \frac{26280 - x}{20435177820774600000} \right ) ^2 = \frac{149598000000 - x}{1.4}.$$

But now I'm stuck.

Could anyone explain to me how to proceed, in order to find $x$?

Answer 1

First, it's useful to write the expression in the standard quadratic form. For ease of reading/writing, I'm going to set $A = 20,435,177,820,744,600,00$ and $B = 149,598,000,000$.

$$ \begin{align*} \left(\frac{26280 - x}{A}\right)^2 &= \frac{B - x}{1.4}\\ \frac{(26280 - x)^2}{A^2} &= \frac{B}{1.4} - \frac{x}{1.4}\\ \frac{26280^2 - 52560x - x^2}{A^2} &= \frac{5}{7}B - \frac{5}{7}x\\ \frac{26280^2}{A^2} - \frac{52560x}{A^2} - \frac{x^2}{A^2} &= \frac{5}{7}B - \frac{5}{7}x\\ 0 &= \frac{x^2}{A^2} + \left(\frac{52560}{A^2} - \frac{5}{7}\right)x + \frac{5}{7}B - \frac{26280^2}{A^2}. \end{align*} $$

You can finish this up with the quadratic formula.

EDIT: Alternatively, there is WolframAlpha.

Answer 2

As Austin Mohr said it's a quadratic equation. You must only expand. Their solutions are: -298283208976198672964919958167422500864 and 149598000000.