Cubic Equation Solver
Solve cubic ax³+bx²+cx+d=0 by Cardano formula. Shows real and complex roots.
Cubic equation: ax³ + bx² + cx + d = 0
A cubic equation ax³ + bx² + cx + d = 0 always has at least one real root, and it can have as many as three. The closed-form answer is Cardano's formula, which appeared in Ars Magna (1545). You start by collapsing the cubic into its depressed form t³ + pt + q = 0 via the substitution x = t − b/(3a). From there, t = ∛(−q/2 + √(q²/4 + p³/27)) + ∛(−q/2 − √(q²/4 + p³/27)). The backstory is messy. Tartaglia worked out the method, handed it over after Cardano swore to keep it quiet, and Cardano printed it regardless. Then there is the awkward part: when the discriminant Δ = q²/4 + p³/27 comes out negative (the casus irreducibilis), the three real roots refuse to be written without taking square roots of negative numbers. That is what pushed mathematicians into their first serious dealings with imaginary numbers, the ones Bombelli would tidy up in 1572.
Applications
Cubics turn up in chemical kinetics (third-order reactions), in solid mechanics (strain-energy and stability problems), in signal processing (cubic transfer functions inside filters), in electronics (the van der Pol oscillator and other cubic nonlinearities), and in thermodynamics (cubic equations of state such as van der Waals). They also show up in exam prep, where the Brazilian ENEM and vestibulares like to hand you a cubic that factors nicely through its rational roots.
FAQ
Can a cubic have no real roots? No. Any cubic with real coefficients has at least one real root, since the function is continuous and runs off to ±∞ at both ends, so it has to cross zero somewhere.
What is the casus irreducibilis? It is the situation where the cubic has three distinct real roots, yet Cardano's formula can only write them using complex numbers. Historically that was the first sign that imaginary numbers couldn't be avoided, even when the answers themselves were perfectly real.
Easier alternative for exam problems? Reach for the rational root theorem. Test the integer divisors of d/a, and once you find one that works, divide by (x − root) to knock the problem down to a quadratic.
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Computes the recommended weight limit from the 1991 revised NIOSH lifting equation, RWL = 23 kg × (25 ÷ H, with H floored at 25) × (1 − 0.003 × |V − 75|) × (0.82 + 4.5 ÷ D) × (1 − 0.0032 × A) × FM × CM, where H is the horizontal distance in centimetres between the hands and the midpoint of the ankles, V is the hand height at the start of the lift, D is the vertical travel of the load and A is the asymmetry angle in degrees. The 23 kg is the load constant, the maximum acceptable under ideal conditions — load against the body, at knuckle height, no trunk twist and lifted infrequently — and each multiplier discounts a fraction as the task departs from that condition. FM, the frequency multiplier, and CM, the coupling multiplier, come from the standard's own tables and therefore enter as data rather than calculation: FM depends on lifts per minute, shift duration and height range; CM on grip quality, rated good, fair or poor. Divide the weight actually lifted by the RWL to get the lifting index: above 1 the task already exposes part of the population to low back risk, and above 3 the risk is high for nearly everyone. Both H and D are floored at 25 cm by the standard itself, so below that the multiplier locks at 1 with no warning on screen: typing the distance in metres instead of centimetres passes validation and returns a limit that is too permissive. Enter the horizontal distance, the starting hand height, the vertical travel, the asymmetry angle, the frequency multiplier and the coupling multiplier.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.