1. The problem is to solve a quadratic equation, which generally has the form $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 2. The formula to find the roots of the quadratic equation is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions for $x$ by calculating the discriminant $\Delta = b^2 - 4ac$. 3. Important rules: - If $\Delta > 0$, there are two distinct real roots. - If $\Delta = 0$, there is exactly one real root (a repeated root). - If $\Delta < 0$, there are two complex roots. 4. To solve a specific quadratic equation, substitute the values of $a$, $b$, and $c$ into the formula and simplify step-by-step. 5. For example, if the equation is $2x^2 - 4x - 6 = 0$: - Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ - Since $\Delta > 0$, there are two real roots. - Calculate the roots: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ - So the roots are: $$x_1 = \frac{4 + 8}{4} = 3$$ $$x_2 = \frac{4 - 8}{4} = -1$$ 6. Therefore, the solutions to the quadratic equation $2x^2 - 4x - 6 = 0$ are $x = 3$ and $x = -1$. This method can be applied to any quadratic equation to find its roots.