Quadratic Equation Ac041C

1. **State the problem:** We are given the equation $$x^2 - x - y = 0$$ and need to understand or manipulate it. 2. **Rewrite the equation:** The equation can be rearranged to express $$y$$ in terms of $$x$$: $$y = x^2 - x$$ 3. **Explanation:** This is a quadratic function in $$x$$, where $$y$$ depends on $$x$$. 4. **Graphical interpretation:** The graph of $$y = x^2 - x$$ is a parabola opening upwards with vertex at $$x = \frac{1}{2}$$. 5. **Vertex calculation:** The vertex $$x$$-coordinate is given by $$x = -\frac{b}{2a}$$ for $$ax^2 + bx + c$$. Here, $$a=1$$, $$b=-1$$, so: $$x = -\frac{-1}{2 \times 1} = \frac{1}{2}$$ 6. **Vertex $$y$$-coordinate:** Substitute $$x=\frac{1}{2}$$ into the equation: $$y = \left(\frac{1}{2}\right)^2 - \frac{1}{2} = \frac{1}{4} - \frac{1}{2} = -\frac{1}{4}$$ 7. **Final vertex:** $$\left(\frac{1}{2}, -\frac{1}{4}\right)$$ This completes the analysis of the given equation.