algebra precalculus - Solving the equation $11x^2-6000x-27500 =0$, preferably without the quadratic formula

I obtained this form while solving an aptitude question.

$$\frac{3000}{x-50} + \frac{3000}{x+50} = 11$$

I changed it into quadratic equation

$$11x^2 -6000x - 27500 =0$$

but I don't know how to solve it.

I can't find two factor for 303500 that sums to 6000 or when I use formula the numbers become huge...
Without using calculator
how to solve it? is there any other simple way to solve [other method]? [or finding factor] I'm a beginner in math. Please explain your answer for me.

Answer

There isn't any standard, guaranteed method apart from the quadratic formula to solve a quadractic equation. However sometimes there are "ad-hoc tricks" which might help you get one root.

The RHS of the equation is an integer; You might suspect that an $x$ such that both the terms on the LHS are integers might be a root (this does not have to be true at all, but it's not bad to try).

Also since $x-50$ and $x+50$ differ by $100$, you want a number $y$ such that both $y$ and $y+100$ divide $3000$. Noticing that $500$ and $600$ satisfy this gives $x=550$ as a root.

Using this, you can find the other root quite easily to be $x=-\frac{50}{11}$ since the product of the roots is $-27500/11$.