Solve any quadratic equation instantly using the quadratic formula. Enter coefficients a, b, and c to get both roots (real or complex), the discriminant, vertex, and axis of symmetry. Shows step-by-step solution. Used by algebra students, teachers, and engineers. Free, instant, no sign-up.
Solve ax² + bx + c = 0 — real & complex roots
The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), solves any quadratic equation of the form ax² + bx + c = 0 where a ≠ 0. The term under the square root, b² − 4ac, is called the discriminant (Δ) and determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (repeated); if Δ < 0, there are two complex conjugate roots (no real solutions). Example: solve x² − 5x + 6 = 0. Here a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1. x = (5 ± √1) / 2 = (5 ± 1) / 2. So x₁ = 3, x₂ = 2. Verify: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓
Quadratic equations appear throughout real-world applications: projectile motion (time to land when thrown), area problems (dimensions of a rectangle given perimeter and area), profit maximization in business (where revenue − cost is a parabola), electrical circuit analysis, and computer graphics (parabolic curves). The quadratic formula was known to Babylonian mathematicians around 2000 BCE, though they solved it geometrically without algebraic notation. The modern symbolic formula was formalized by Hindu mathematician Brahmagupta in 628 CE and refined in Europe during the 16th century.
Δ = b² − 4ac tells you everything about the solutions before you solve. Δ > 0: two different real roots (parabola crosses x-axis twice). Δ = 0: one real root, the vertex touches the x-axis. Δ < 0: no real solutions — complex (imaginary) roots. Δ is a perfect square: roots are rational. Quick check: for x² + 2x + 5 = 0, Δ = 4 − 20 = −16 < 0 → no real roots.
The vertex of the parabola y = ax² + bx + c is at (h, k) where h = −b/(2a) and k = c − b²/(4a). The vertex is the minimum point if a > 0 (opens upward) or maximum point if a < 0 (opens downward). For x² − 5x + 6: h = 5/2 = 2.5, k = 6 − 25/4 = −0.25. Vertex = (2.5, −0.25).
Factoring is faster when roots are integers: x² − 5x + 6 = (x−2)(x−3). But factoring only works for "nice" integer roots. The quadratic formula always works for any coefficients. Completing the square works too but is more steps. Use factoring first (mental math check); use the quadratic formula when factoring isn't obvious or roots are fractions/decimals.
A ball is thrown upward at 64 ft/sec from a 6 ft height. Height = −16t² + 64t + 6. When does it land? Set = 0: −16t² + 64t + 6 = 0, or 16t² − 64t − 6 = 0. a=16, b=−64, c=−6. Δ = 4096 + 384 = 4480. t = (64 ± √4480)/32. t ≈ (64 + 66.93)/32 ≈ 4.09 seconds to land.