Complete guide to solving quadratic equations using the quadratic formula, factoring, and completing the square. Learn each method with step-by-step examples and real-world applications in physics and business.
What Is a Quadratic Equation?
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It's called "quadratic" from the Latin "quadratus" (square) because the variable appears squared.
Method 1: The Quadratic Formula
x = [−b ± √(b² − 4ac)] / 2a. Example: x² − 5x + 6 = 0. a=1, b=−5, c=6. Discriminant: 25 − 24 = 1. x = [5 ± 1] / 2 → x = 3 or x = 2.
Method 2: Factoring
Find two numbers that multiply to c and add to b. x² − 5x + 6: numbers are −2 and −3. (x − 2)(x − 3) = 0 → x = 2 or x = 3. Best for simple equations with integer roots.
Method 3: Completing the Square
Transform ax² + bx + c = 0 into (x + p)² = q. Useful when factoring isn't obvious and you want to understand the graph's vertex form.
The Discriminant
The discriminant (b² − 4ac) tells you the number and type of solutions: Positive → 2 real solutions. Zero → 1 real solution (double root). Negative → 0 real solutions (2 complex solutions).
Real-World Applications
Physics: Projectile motion (h = −4.9t² + v₀t). Quadratic equations describe the path of any object under gravity. Business: Profit maximization — revenue and cost are often quadratic functions. Engineering: Circuit design uses quadratic equations for component values.
Key Takeaways
- Quadratic formula works for every quadratic equation — memorize it
- Factoring is faster for simple equations — look for integer pairs that multiply to c and add to b
- The discriminant tells you how many real solutions exist
- Always check your solutions by substituting back into the original equation
Ready to solve? Use our Quadratic Solver for instant step-by-step solutions.