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 $x^2 − x − α = 0$. What is the value of $α$?

Forty-seventh Annual Columbus State Invitational Mathematics Tournament

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