1. The problem is to solve a quadratic equation of the form $ax^2 + bx + c = 0$ using the quadratic formula. 2. The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula finds the roots (solutions) of any 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 use the formula, identify $a$, $b$, and $c$ from your quadratic equation. 5. Substitute these values into the formula and simplify step-by-step: - Calculate the discriminant $D = b^2 - 4ac$. - Find the square root $\sqrt{D}$. - Compute the numerator $-b \pm \sqrt{D}$. - Divide by $2a$ to get the roots. 6. Example: Solve $2x^2 - 4x - 6 = 0$. - Here, $a=2$, $b=-4$, $c=-6$. - Discriminant: $D = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64$. - Square root: $\sqrt{64} = 8$. - Roots: $x = \frac{-(-4) \pm 8}{2 \times 2} = \frac{4 \pm 8}{4}$. - So, $x_1 = \frac{4 + 8}{4} = 3$, and $x_2 = \frac{4 - 8}{4} = -1$. 7. Therefore, the solutions to the quadratic equation are $x=3$ and $x=-1$.