1. Let's start by stating the problem: solving quadratic simultaneous equations means finding values of variables that satisfy two equations, where at least one is quadratic. 2. A common form is: $$\begin{cases} ax^2 + bx + c = 0 \\ dx + ey + f = 0 \end{cases}$$ 3. Important rules: - Quadratic equations involve terms like $x^2$. - Simultaneous means both equations must be true at the same time. 4. To solve, use substitution or elimination: - Solve one equation for one variable. - Substitute into the other equation. 5. Example: $$\begin{cases} x^2 + y = 7 \\ x + y = 3 \end{cases}$$ 6. From the second equation, express $y$: $$y = 3 - x$$ 7. Substitute into the first: $$x^2 + (3 - x) = 7$$ 8. Simplify: $$x^2 - x + 3 = 7$$ 9. Move all terms to one side: $$x^2 - x + 3 - 7 = 0$$ $$x^2 - x - 4 = 0$$ 10. Solve quadratic using the formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, $c=-4$. 11. Calculate discriminant: $$\Delta = (-1)^2 - 4 \times 1 \times (-4) = 1 + 16 = 17$$ 12. Find roots: $$x = \frac{1 \pm \sqrt{17}}{2}$$ 13. Find corresponding $y$ values: $$y = 3 - x$$ 14. So solutions are: $$\left( \frac{1 + \sqrt{17}}{2}, 3 - \frac{1 + \sqrt{17}}{2} \right) \text{ and } \left( \frac{1 - \sqrt{17}}{2}, 3 - \frac{1 - \sqrt{17}}{2} \right)$$ This method works for any quadratic simultaneous equations by substitution and solving the resulting quadratic.