1. Let's start by 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. The formula used to solve quadratic equations is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions (roots) of the quadratic equation. 3. Important rules: - The term under the square root, $b^2 - 4ac$, is called the discriminant. - If the discriminant is positive, there are two distinct real roots. - If it is zero, there is exactly one real root (a repeated root). - If it is negative, the roots are complex (not real). 4. Let's see an example: Solve $$2x^2 - 4x - 6 = 0$$. 5. Identify coefficients: $a=2$, $b=-4$, $c=-6$. 6. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 7. Since $\Delta = 64 > 0$, there are two real roots. 8. Apply the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 9. Calculate each root: - $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ - $$x_2 = \frac{4 - 8}{4} = \frac{-4}{4} = -1$$ 10. Final answer: The solutions to the quadratic equation are $$x = 3$$ and $$x = -1$$. This process can be used to solve any quadratic equation by identifying $a$, $b$, and $c$, computing the discriminant, and applying the quadratic formula.