1. The problem is to understand and use the quadratic formula to find the roots of a quadratic equation $ax^2 + bx + c = 0$. 2. The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula calculates the values of $x$ that satisfy 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 real roots. - If it is zero, there is one real root (a repeated root). - If it is negative, the roots are complex (not real). 4. To solve a quadratic equation using this formula: - Identify $a$, $b$, and $c$ from the equation. - Calculate the discriminant $D = b^2 - 4ac$. - Substitute $a$, $b$, and $D$ into the formula. - Simplify to find the roots. 5. Example: Solve $2x^2 + 3x - 2 = 0$. - Here, $a=2$, $b=3$, $c=-2$. - Calculate discriminant: $D = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$. - Substitute into formula: $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ - Calculate roots: $$x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{\cancel{2}}{\cancel{4}} = \frac{1}{2}$$ $$x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = \frac{\cancel{-8}}{\cancel{4}} = -2$$ 6. Final answer: The roots are $x = \frac{1}{2}$ and $x = -2$.