1. **Stating the problem:** 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 $a \neq 0$. The goal is to find the values of $x$ that make the equation true. 2. **Formula used:** The most common way to solve quadratic equations is by using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula gives the solutions (roots) of the quadratic equation. 3. **Important rules:** - The expression under the square root, $b^2 - 4ac$, is called the discriminant. - 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). 4. **Example 1:** Solve $$2x^2 + 3x - 2 = 0$$ - Identify $a=2$, $b=3$, $c=-2$. - Calculate the discriminant: $$3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25$$ - Since 25 is positive, there are two real solutions. - Apply the quadratic formula: $$x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4}$$ - Calculate each solution: $$x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5$$ $$x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$ 5. **Example 2:** Solve $$x^2 - 4x + 4 = 0$$ - Identify $a=1$, $b=-4$, $c=4$. - Calculate the discriminant: $$(-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$$ - Since the discriminant is zero, there is one real solution. - Apply the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{0}}{2 \times 1} = \frac{4}{2} = 2$$ 6. **Summary:** The quadratic formula is a powerful tool to solve any quadratic equation by plugging in the values of $a$, $b$, and $c$. Understanding the discriminant helps predict the number and type of solutions. This explanation is designed to be clear and simple enough for a 15-year-old to understand.