Quadratic equations are one of the most fundamental topics in algebra. An equation of the form ax² + bx + c = 0, where a ≠ 0, is called a quadratic equation. In this guide, we'll explore three main methods to solve them.
Method 1: The Discriminant Formula. The most universal method uses the discriminant D = b² - 4ac. If D > 0, there are two distinct real roots: x = (-b ± √D) / 2a. If D = 0, there is exactly one root (a repeated root). If D < 0, there are no real roots — only complex ones.
Method 2: Factoring. When the quadratic can be factored into (x - r₁)(x - r₂) = 0, the roots are simply r₁ and r₂. For example, x² - 5x + 6 = (x - 2)(x - 3) = 0 gives us x = 2 and x = 3. This method is fast but not always applicable.
Method 3: Completing the Square. This technique rewrites ax² + bx + c as a(x - h)² + k, making it easy to solve. Start by dividing by a, then add and subtract (b/2a)² inside the equation. This method is especially useful for deriving the quadratic formula itself.
Practice tip: Always check your answers by substituting them back into the original equation. This simple verification step helps catch arithmetic errors and builds mathematical intuition. With regular practice, you'll be solving quadratic equations quickly and confidently.