1. The problem is to solve an equation or expression using an alternative method. 2. Without a specific equation given, let's consider a common example: solving the quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula. 3. The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ which gives the roots of any quadratic equation. 4. Important rules: - The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots. - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, one real root (repeated). - If $\Delta < 0$, two complex roots. 5. Example: Solve $2x^2 - 4x - 6 = 0$ using the quadratic formula. 6. Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$$ 7. Since $\Delta = 64 > 0$, there are two real roots. 8. Substitute into the formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 9. Calculate each root: $$x_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3$$ $$x_2 = \frac{4 - 8}{4} = \frac{\cancel{4} - 8}{\cancel{4}} = \frac{-4}{1} = -1$$ 10. Final answer: The roots are $x = 3$ and $x = -1$. This method is an alternative to factoring or completing the square and works for all quadratic equations.