Quadratic Equation Solver
Solve ax² + bx + c = 0 — real and complex roots, vertex, discriminant.
The Quadratic Formula
The quadratic formula solves any equation of the form ax² + bx + c = 0:
x = ( −b ± √(b² − 4ac) ) / (2a)
It works for any quadratic, even when factoring isn't obvious or possible. The formula was known to Babylonians as early as 2000 BCE in geometric form, and reached its modern algebraic form via medieval Arabic mathematicians (Al-Khwarizmi, ~820 CE). The Wikipedia quadratic formula article covers derivation by completing the square.
The Discriminant Tells You What to Expect
The expression D = b² − 4ac (called the discriminant) reveals the nature of the solutions before you compute them:
| Discriminant | Solutions | Graph Behavior |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| D = 0 | One repeated real root (double root) | Parabola touches x-axis at vertex |
| D < 0 | Two complex conjugate roots | Parabola doesn't touch x-axis |
How long until the ball hits the ground?
A ball is thrown up at 20 m/s from a height of 5 m. Gravity pulls it down at 9.8 m/s². Its height over time follows:
h(t) = −4.9t² + 20t + 5
Setting h(t) = 0 (ground) gives a quadratic with a = −4.9, b = 20, c = 5. Discriminant: D = 400 − 4(−4.9)(5) = 498. Roots:
t = (−20 ± √498) / (−9.8) ≈ 4.31 s or −0.24 s
The negative root is physically meaningless (time before the throw). The ball hits the ground at ~4.31 seconds. This is the canonical engineering use of quadratics — modeling parabolic motion, structural arches, optimization problems.
Vertex Form and the Parabola
The vertex of y = ax² + bx + c is at:
x = −b / (2a) , y = c − b² / (4a)
If a > 0, the parabola opens upward — vertex is the minimum point. If a < 0, it opens downward — vertex is the maximum. This is critical for optimization problems (maximize area, profit, etc.).
Where Quadratics Show Up in Real Life
- Physics: projectile motion, springs, simple harmonic motion
- Engineering: structural arches, satellite dishes, suspension bridges
- Economics: revenue maximization, supply and demand intersections
- Computer graphics: Bézier curves, parabolic interpolation
- Finance: Black-Scholes option pricing (in part)