1. The problem is to understand and use examples effectively in math problems. 2. Examples help illustrate how to apply formulas and solve problems step-by-step. 3. For instance, if we want to solve a quadratic equation $ax^2 + bx + c = 0$, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. Important rules include checking the discriminant $\Delta = b^2 - 4ac$ to determine the nature of roots. 5. If $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, one real root; if $\Delta < 0$, complex roots. 6. Using an example: Solve $2x^2 - 4x - 6 = 0$. 7. Calculate discriminant: $\Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$. 8. Since $\Delta > 0$, two real roots exist. 9. Apply formula: $$x = \frac{-(-4) \pm \sqrt{64}}{2 \times 2} = \frac{4 \pm 8}{4}$$ 10. Roots are $x = \frac{4 + 8}{4} = 3$ and $x = \frac{4 - 8}{4} = -1$. 11. This example shows how to use the quadratic formula and interpret the discriminant.