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.

Multiply by 11, and replace $y=11x$. Then you get
$$y^2-6000y-302500=0 \,.$$
Now complete the square:
$$y^2-6000y+3000^2=3000^2+302500$$
Last:
$$3000^2+302500=3000\times 3000+3025\times 100=600 \times 5 \times 6 \times 500+121\times25\times100$$
$$=2500 \times (3600+121)=2500 \times 3721=50^2 \times 61^2$$
Thus you get
$$(y+3000)^2=3050^2$$
