Cubic Equation Solver (ax³ + bx² + cx + d = 0)
The quadratic formula solves any equation ax² + bx + c = 0. The discriminant b²−4ac tells you how many real solutions exist before you solve.
- 1Write in standard form: ax² + bx + c = 0
- 2Calculate discriminant: Δ = b² − 4ac
- 3Δ > 0: two real roots; Δ = 0: one root; Δ < 0: complex roots
- 4Apply: x = (−b ± √Δ) / 2a
x²−5x+6=0=x=3 or x=2Δ=1>0, two roots
x²−2x+1=0=x=1 (repeated)Δ=0, one root
x²+x+1=0=Complex rootsΔ=−3<0
| Δ = b²−4ac | Real roots | Parabola |
|---|---|---|
| Δ > 0 | Two distinct | Crosses x-axis twice |
| Δ = 0 | One repeated | Touches x-axis once |
| Δ < 0 | None (complex) | Never crosses |
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Fun Fact
The quadratic formula was known to Babylonian mathematicians c. 2000 BC, though written in prose form, not modern algebra notation.
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