1. Let's start by stating the problem: You want to understand what a quadratic equation is and how to solve it. 2. A quadratic equation is any equation that can be written in the form $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are numbers and $a \neq 0$. 3. The most important formula to solve quadratic equations is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions (roots) of the quadratic equation. 4. Let's explain the parts: - $b^2 - 4ac$ is called the discriminant. It tells us about the nature of the roots. - If the discriminant is positive, there are two real solutions. - If it is zero, there is exactly one real solution. - If it is negative, there are no real solutions (but two complex solutions). 5. To solve a quadratic equation, plug the values of $a$, $b$, and $c$ into the formula and simplify step-by-step. 6. For example, if we have $$2x^2 + 3x - 2 = 0$$ then $a=2$, $b=3$, and $c=-2$. 7. Calculate the discriminant: $$b^2 - 4ac = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$$ 8. Since the discriminant is positive, there are two real solutions. 9. Apply the quadratic formula: $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ 10. Calculate each solution: $$x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{\cancel{2}}{\cancel{4}} = \frac{1}{2}$$ $$x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ 11. So the solutions are $x = \frac{1}{2}$ and $x = -2$. This is how you solve quadratic equations using the quadratic formula. Practice with different values to get comfortable!