1. **State the problem:** Given the system of equations: $$x + y = 6$$ $$xy = 36$$ Find the values of $x$ and $y$. 2. **Use the quadratic equation approach:** We know $x$ and $y$ satisfy the quadratic equation whose roots are $x$ and $y$: $$t^2 - (x+y)t + xy = 0$$ Substitute the given sums and products: $$t^2 - 6t + 36 = 0$$ 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-6)^2 - 4 \times 1 \times 36 = 36 - 144 = -108$$ 4. **Interpret the discriminant:** Since $\Delta < 0$, the roots are complex (no real solutions). 5. **Find the complex roots:** $$t = \frac{6 \pm \sqrt{-108}}{2} = \frac{6 \pm i\sqrt{108}}{2} = 3 \pm i3\sqrt{3}$$ 6. **Final answer:** $$x = 3 + 3i\sqrt{3}, \quad y = 3 - 3i\sqrt{3}$$ These are the values of $x$ and $y$ that satisfy the system.