Solve System F95D11

1. **State the problem:** We are given the system of equations: $$x + y = 4$$ $$xy = 16$$ We want to find the values of $x$ and $y$ that satisfy both equations. 2. **Use the formulas and rules:** From the first equation, we know the sum of $x$ and $y$ is 4. From the second, their product is 16. 3. **Express as a quadratic equation:** Let’s consider $x$ and $y$ as roots of a quadratic equation. The quadratic with roots $x$ and $y$ is: $$t^2 - (x+y)t + xy = 0$$ Substitute the known values: $$t^2 - 4t + 16 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$t = \frac{4 \pm \sqrt{4^2 - 4 \times 1 \times 16}}{2 \times 1} = \frac{4 \pm \sqrt{16 - 64}}{2} = \frac{4 \pm \sqrt{-48}}{2}$$ 5. **Simplify the discriminant:** $$\sqrt{-48} = \sqrt{-1 \times 16 \times 3} = 4i\sqrt{3}$$ 6. **Write the solutions:** $$t = \frac{4 \pm 4i\sqrt{3}}{2} = 2 \pm 2i\sqrt{3}$$ 7. **Final answer:** The solutions for $x$ and $y$ are complex conjugates: $$x = 2 + 2i\sqrt{3}, \quad y = 2 - 2i\sqrt{3}$$ or vice versa.