The Power of Parabolas: A Guide to the Quadratic Formula

The quadratic equation, an equation of the form ax² + bx + c = 0, is a cornerstone of algebra. Its graph is a symmetric curve called a parabola, and its solutions, or 'roots', represent where this parabola intersects the x-axis. These equations appear in countless real-world scenarios, from calculating the trajectory of a projectile in physics to optimizing profit margins in economics and modeling suspension bridge cables in engineering.

While some quadratics can be solved by factoring, the quadratic formula works for *all* quadratic equations. This calculator automates the process, instantly giving you the roots along with the discriminant (Δ = b² - 4ac), which determines whether the roots are real or complex.

The Quadratic Formula

For any quadratic equation ax² + bx + c = 0 (where a ≠ 0):

x = [-b ± √(b² - 4ac)] / 2a

Real-World Uses

• Physics: Projectile motion, rocket trajectories.
• Engineering: Suspension bridges, parabolic dishes.
• Economics: Profit optimization, revenue models.
• Geometry: Solving area/dimension problems.