Cubic Equation
Solve cubic equations of the form ax^3 + bx^2 + cx + d = 0 using Cardano's method. Get real and complex roots instantly for any cubic polynomial.
About This Calculator
The Cubic Equation Calculator solves polynomial equations of degree three in the form ax^3 + bx^2 + cx + d = 0 (where a is non-zero). Using Cardano's method, it computes all three roots -- including real roots, repeated roots, and complex conjugate pairs.
Enter the four coefficients a, b, c, and d, and the calculator applies the depressed cubic transformation t^3 + pt + q = 0 to find the roots analytically. The discriminant reveals the nature of the roots: three distinct real roots, one real root with two complex conjugates, or multiple real roots.
This tool is ideal for students studying algebra, engineers working with cubic polynomials, and anyone who needs to quickly solve cubic equations for academic or professional work.
Frequently Asked Questions
What is a cubic equation?
A cubic equation is a polynomial equation of degree three with the general form ax^3 + bx^2 + cx + d = 0, where a ≠ 0. Every cubic equation has exactly three roots (solutions) in the complex number system, though some may be repeated or complex.
How does the Cubic Equation Calculator work?
The calculator uses Cardano's method to solve the cubic equation. It first depresses the cubic by substituting x = t - b/(3a), then computes the discriminant to determine whether the roots are real or complex. The discriminant tells us whether we have one real root and two complex conjugates (positive), three distinct real roots (negative), or repeated roots (zero).
Can the coefficient a be zero?
No. If a = 0, the equation reduces to a quadratic (degree 2) and is no longer a cubic equation. The calculator requires a non-zero value for the coefficient a.
How do I interpret complex roots?
Complex roots appear in conjugate pairs in the form a + bi and a - bi, where i = sqrt(-1). If the discriminant is positive, one root is real and the other two are complex conjugates. Complex roots indicate the cubic polynomial crosses the x-axis only once.
What is the discriminant of a cubic equation?
The discriminant Delta = (q/2)^2 + (p/3)^3 determines the nature of the roots, where p and q come from the depressed cubic. If Delta > 0, one root is real and two are complex. If Delta = 0, all roots are real with at least one repeated. If Delta < 0, all three roots are real and distinct.
Is this cubic equation solver free?
Yes, this cubic equation calculator is completely free to use with no limits on the number of calculations. It works for any real coefficients, and results include both real and complex roots.