1. The problem asks to solve the quadratic equation graphically for the first equation: $x^2 - 3x - 3 = 0$. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. 3. To solve graphically, we consider the function $y = x^2 - 3x - 3$ and find the points where the graph intersects the x-axis (i.e., where $y=0$). 4. The roots of the equation are the x-values where the parabola crosses the x-axis. 5. We can find the roots algebraically using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-3$, and $c=-3$. 6. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-3)^2 - 4(1)(-3) = 9 + 12 = 21$$ 7. Since $\Delta > 0$, there are two distinct real roots. 8. Calculate the roots: $$x = \frac{-(-3) \pm \sqrt{21}}{2(1)} = \frac{3 \pm \sqrt{21}}{2}$$ 9. The approximate decimal values are: $$x_1 = \frac{3 + 4.58}{2} = \frac{7.58}{2} = 3.79$$ $$x_2 = \frac{3 - 4.58}{2} = \frac{-1.58}{2} = -0.79$$ 10. These are the x-intercepts of the graph of $y = x^2 - 3x - 3$. Final answer: The solutions to $x^2 - 3x - 3 = 0$ are $$x = \frac{3 \pm \sqrt{21}}{2}$$ or approximately $x = 3.79$ and $x = -0.79$.