1. The problem is to find when a quadratic expression becomes zero, which means solving for the roots of the quadratic equation. 2. A quadratic equation is generally written as $ax^2 + bx + c = 0$. 3. To find the roots, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. The term under the square root, $b^2 - 4ac$, is called the discriminant and determines the nature of the roots. 5. If the discriminant is zero, the quadratic has exactly one real root (a repeated root). 6. If the discriminant is positive, there are two distinct real roots. 7. If the discriminant is negative, the roots are complex (not real). 8. To find when the quadratic becomes zero, set the quadratic equal to zero and solve using the formula above. 9. For example, if the quadratic is $x^2 - 3x + 2 = 0$, then $a=1$, $b=-3$, and $c=2$. 10. Calculate the discriminant: $$b^2 - 4ac = (-3)^2 - 4 \times 1 \times 2 = 9 - 8 = 1$$ 11. Since the discriminant is positive, there are two real roots. 12. Calculate the roots: $$x = \frac{-(-3) \pm \sqrt{1}}{2 \times 1} = \frac{3 \pm 1}{2}$$ 13. So the roots are: $$x_1 = \frac{3 + 1}{2} = 2$$ $$x_2 = \frac{3 - 1}{2} = 1$$ 14. Therefore, the quadratic becomes zero at $x=1$ and $x=2$. 15. This means the quadratic crosses the x-axis at these points, which are the solutions to the equation.