Follow one set of numbers
x⁴−5x²+4=(x²−1)(x²−4), giving roots −2,−1,1,2.
Math calculator
Quartic Equation Solver
Find all four real or complex roots of ax⁴ + bx³ + cx² + dx + e = 0 numerically. A checkable formula accompanies four roots instead of leaving an unexplained number.
Quartic Equation Solver inputs
Known quantities
Problems this can answer
Quartics arise from geometric intersections, energy models, optimization, and characteristic equations.
The idea behind the result
A quartic has four complex roots counting multiplicity. With real coefficients, every nonreal root is paired with its complex conjugate.
The calculation treats quartic coefficient a, cubic coefficient b, quadratic coefficient c, linear coefficient d and constant e according to their labels, not as interchangeable values in quartic equation solver.
Reproducing the answer
Normalize coefficients, refine four complex estimates simultaneously, cluster near-equal values, and verify small polynomial residuals. To continue from Quartic Equation Solver, try rebuild from factors.
Mistakes worth catching
The leading coefficient must be nonzero. Closely repeated roots are numerically delicate, so displayed values should be checked in the original polynomial. If the Quartic Equation Solver assumptions do not fit, consider verify each root.
Changing only one field helps distinguish a data-entry problem from the intended behavior of quartic equation solver.
How the inputs shape the output
A field name is part of the formula. Match the problem's quantities to quartic coefficient a, cubic coefficient b, quadratic coefficient c, linear coefficient d and constant e, then check that they share the scale assumed by the quartic equation solver relationship.
How the result responds
Changing the constant shifts the graph vertically and can alter the number of real crossings by pairs. This is a stronger check than judging the answer only by how many decimal places it shows.
A quartic solver finds every root; completing the square applies only to quadratic expressions. The formula panel makes the chosen definition explicit.
Recording the answer
Match the reported precision to quartic coefficient a, cubic coefficient b, quadratic coefficient c, linear coefficient d and constant e, not to the number of digits the browser can display. Preserve an exact form when it communicates the quartic equation solver structure more clearly than a decimal.
The minimum audit trail is short: quartic coefficient a, cubic coefficient b, quadratic coefficient c, linear coefficient d and constant e, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Questions about Quartic Equation Solver
Can all four roots be real?
Yes.
Can there be no real roots?
Yes.
Are complex roots valid solutions?
Yes, over the complex numbers.
Why might a repeated root be sensitive?
Small coefficient errors can split it into nearby roots.