1. **Problem Statement:** We want to understand how to solve systems of quadratic equations, which are sets of two or more equations where at least one equation is quadratic (degree 2). 2. **General Form:** A quadratic equation in two variables $x$ and $y$ looks like $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $a,b,c,d,e,f$ are constants. 3. **Common Methods to Solve Systems of Quadratic Equations:** - **Substitution:** Solve one equation for one variable and substitute into the other. - **Elimination:** Add or subtract equations to eliminate one variable. - **Graphical:** Find intersection points of the curves represented by the equations. 4. **Example:** Solve the system $$\begin{cases} y = x^2 + 1 \\ y = 3x + 5 \end{cases}$$ 5. **Step 1: Set the equations equal since both equal $y$:** $$x^2 + 1 = 3x + 5$$ 6. **Step 2: Rearrange to form a quadratic equation:** $$x^2 - 3x + 1 - 5 = 0 \Rightarrow x^2 - 3x - 4 = 0$$ 7. **Step 3: Factor or use quadratic formula:** Factors of $-4$ that sum to $-3$ are $-4$ and $1$, so $$(x - 4)(x + 1) = 0$$ 8. **Step 4: Solve for $x$:** $$x = 4 \quad \text{or} \quad x = -1$$ 9. **Step 5: Find corresponding $y$ values using $y = 3x + 5$:** - For $x=4$: $y = 3(4) + 5 = 12 + 5 = 17$ - For $x=-1$: $y = 3(-1) + 5 = -3 + 5 = 2$ 10. **Solutions:** $$(4, 17) \quad \text{and} \quad (-1, 2)$$ 11. **Shortcuts for Multiple Choice:** - Substitute answer choices into the equations to check quickly. - Use symmetry or special values (like $x=0$) to test. - Look for factoring opportunities to avoid quadratic formula. 12. **Summary:** To solve systems of quadratic equations, isolate variables, substitute, and solve resulting quadratics. Practice factoring and substitution to speed up solving multiple choice problems.