1. **Stating the problem:** A quadratic equation is any equation that can be written in the form $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. **Methods to solve quadratic equations:** - **Factoring:** Express the quadratic as a product of two binomials, then set each equal to zero. - **Quadratic formula:** Use $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ which works for all quadratics. - **Completing the square:** Rewrite the equation to form a perfect square trinomial. - **Graphing:** Find the points where the parabola crosses the x-axis. 3. **Step-by-step example using the quadratic formula:** Given $$2x^2 - 4x - 6 = 0$$ - Identify coefficients: $a=2$, $b=-4$, $c=-6$. - Calculate the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$. - Since $\Delta > 0$, there are two real solutions. - Apply the formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ - Calculate each root: - $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ - $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 4. **Common mistakes to avoid:** - Forgetting to set the equation equal to zero before solving. - Miscalculating the discriminant or forgetting the $\pm$ sign in the quadratic formula. - Not simplifying the square root fully. - Errors in factoring by missing factors or signs. 5. **Summary:** Choose the method based on the equation's form. Factoring is quickest if possible, otherwise use the quadratic formula for a guaranteed solution. Always check your solutions by plugging them back into the original equation.