Quadratic Formula Guide: Solve ax² + bx + c = 0 With Examples & the Discriminant
x = (−b ± √(b²−4ac)) / 2a. The quadratic formula finds both roots of any quadratic equation. Learn the discriminant shortcut that reveals the solution type before calculating.
The Quadratic Formula
The quadratic formula is derived from completing the square on the general form ax² + bx + c = 0. It gives the exact values of x where a parabola (the graph of a quadratic function) crosses the x-axis — the roots of the equation. The ± symbol means the formula produces two solutions: one using addition and one using subtraction. Both must be calcul
The Discriminant: D = b² − 4ac
The discriminant D = b² − 4ac tells you the nature of the solutions before you complete the calculation — a valuable shortcut in tests and real-world analysis. D > 0: two distinct real roots (the parabola intersects the x-axis at two points). D = 0: exactly one real root (a repeated root — the parabola just touches the x-axis at its vertex). D tria
Worked Examples
Example 1: 2x² − 7x + 3 = 0. D = 49 − 24 = 25. x = (7 ± 5) / 4 → x = 3 or x = 0.5. Check: 2(9) − 21 + 3 = 0 ✓; 2(0.25) − 3.5 + 3 = 0 ✓ Example 2: x² + 4x + 4 = 0. D = 16 − 16 = 0. x = −4/2 = −2. One repeated root. Check: 4 − 8 + 4 = 0 ✓ Example 3: x² + x + 1 = 0. D = 1 − 4 = −3 discriminant calculator to run the calculation for your scenario.
Real-World Applications
Quadratic equations appear naturally whenever a relationship involves a squared variable. Common applications: Projectile motion: h(t) = h₀ + v₀t − 4.9t² (metric). When does the object hit the ground? Set h = 0, solve for t using the quadratic formula. Take only the positive root. Area problems: a rectangular garden has perimeter 48m and area 128m²
Frequently Asked Questions
When should I use the quadratic formula vs factoring?
Use factoring when the discriminant is a perfect square and coefficients are integers — it's faster. Use the quadratic formula when: the discriminant is not a perfect square, coefficients are decimals or fractions, or when you need exact irrational roots. The quadratic formula al
How do I remember the quadratic formula?
The most common memorization technique is singing it to the tune of 'Pop Goes the Weasel': 'x equals negative b, plus or minus square root, of b squared minus four a c, all over two a.' Another approach: derive it by completing the square on ax² + bx + c = 0 — practicing the deri
What is a quadratic equation used for in real life?
Quadratic equations model any phenomenon involving squared variables: projectile trajectories (ballistics, sports, engineering), the area of geometric shapes, profit/cost/revenue relationships in business, the motion of pendulums, lens and mirror focusing in optics, and voltage/c
What do complex roots mean geometrically?
When D 0, the entire parabola sits above the x-axis (the expression is always positive). If a < 0, it's always negative. Complex roots have no x-intercept interpretation but have important applications in electrical engineering (AC circuit analysis) and control systems.
How is completing the square related to the quadratic formula?
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. Starting with ax² + bx = −c, dividing by a, adding (b/2a)² to both sides, the left side becomes (x + b/2a)² and the right side becomes (b² − 4ac)/4a². Taking the square root and solvin