What is the value of α?

In the accompaning figure the coordinates of the triangle $ABC$ are $A=(-1,1)$, $B=(1,1)$ and $C=(1,0)$. 

We denote the slope $m$ of the line $LM$ which passes through the origin of the axis, the point $O$, and divides the area of the triangle $ABC$ in half. The equation that m satisfies is the quadratic $x^2-x-\alpha =0$. What is the value of $\alpha$?

Forty-seventh Annual Columbus State Invitational Mathematics Tournament

Κάντε κλικ εδώ, για να δείτε τη λύση που μου έστειλε ο κ. Κώστας Δόρτσιος.