Solve cubic equations of the form ax³ + bx² + cx + d = 0, set decimal precision, and plot the graph.
ax³ + bx² + cx + d = 0
This calculator solves cubic equations of the form ax³ + bx² + cx + d = 0 using Cardano’s Method. It calculates all real and complex roots with adjustable precision and plots the cubic graph instantly for better visualization.
Enter the four coefficients:
The calculator automatically computes all three roots of the cubic equation.
You can control the number of decimal places:
This ensures accuracy based on your required level of detail.
The calculator plots the cubic function:
This helps you understand how the roots relate to the curve.
The calculator updates instantly:
This cubic equation calculator automatically determines the roots of an equation of the form ax³ + bx² + cx + d = 0. The tool follows a mathematical process based on Cardano’s method, which is a classical algebra technique used to solve third-degree polynomial equations. Below is a simplified explanation of the steps used by the calculator.
Step 1 – Enter the coefficients
The calculator requires four numerical inputs:
These coefficients define the cubic polynomial that will be solved.
Step 2 – Normalize the equation
The calculator divides the equation by a to simplify the expression and
prepare it for Cardano’s method. This produces a normalized form that makes
further calculations easier.
Step 3 – Convert to a depressed cubic
A substitution is applied to remove the squared term of the equation.
The transformation used is:
x = t − (b / 3a)
After this substitution, the equation becomes a simplified cubic known as a depressed cubic. The calculator then computes two important parameters:
These parameters are used to determine the nature of the roots.
Step 4 – Compute the discriminant
The discriminant helps determine how many real or complex solutions the equation has.
Based on this result, the calculator selects the appropriate formula from Cardano’s method to compute all roots of the cubic equation.
Step 5 – Calculate and display the roots
The solutions are calculated and then rounded according to the
decimal precision selected by the user. The calculator displays
all real and complex roots clearly in the result section.
Step 6 – Plot the cubic graph
After solving the equation, the calculator also plots the function
y = ax³ + bx² + cx + d. The graph shows how the cubic curve behaves
across the selected range of x-values and visually confirms where the
curve intersects the x-axis (the roots).
This automated process allows the calculator to quickly solve cubic equations while also helping users understand how the polynomial behaves graphically.
A cubic equation is a polynomial equation of degree three. It contains a variable raised to the third power and can produce up to three solutions called roots. In algebra, the standard form of a cubic equation is widely used to analyze curves, model physical systems, and solve mathematical problems involving nonlinear relationships.
ax³ + bx² + cx + d = 0
In this equation:
To solve cubic equations analytically, mathematicians use Cardano’s Method, a classical algebraic technique developed in the 16th century. This method transforms the original cubic equation into a simplified form that makes it easier to compute the roots. Depending on the values of the coefficients, a cubic equation may have:
Because cubic equations can behave differently depending on their coefficients, graphing the function is often useful. The cubic curve can cross the x-axis up to three times, which visually represents the roots of the equation.
Understanding the cubic equation formula helps students, engineers, and researchers analyze nonlinear systems, polynomial models, and mathematical relationships that cannot be represented by simple linear or quadratic equations.
Consider the equation:
x³ - 6x² + 11x - 6 = 0
The calculator computes:
Roots: 1, 2, 3
The graph shows the curve intersecting the x-axis at x = 1, 2, and 3, confirming the three real solutions.
Everything you need to know about solving cubic equations and plotting their graphs online.
This calculator solves cubic equations of the form ax³ + bx² + cx + d = 0 using Cardano’s method. It computes all three roots (real or complex) and also plots the corresponding cubic graph.
You must enter numerical values for coefficients a (x³), b (x²), c (x), and d (constant). The calculator automatically computes the roots once valid values are provided.
If a = 0, the equation is no longer cubic. The calculator will display an error message because Cardano’s method only applies to true cubic equations.
Yes. If the discriminant is positive, the calculator returns one real root and two complex conjugate roots in the form a ± bi. All roots are displayed clearly.
When the discriminant is negative, the equation has three distinct real roots. The calculator computes all three using trigonometric formulas derived from Cardano’s method.
The decimal precision field allows you to choose how many decimal places the roots are rounded to. You can set precision between 0 and 10 decimal places.
The calculator plots the function y = ax³ + bx² + cx + d using a dynamic line chart. The graph updates automatically when you change coefficients or adjust the x-range.
Yes. You can enter custom initial and final x-values for plotting. If left empty, the default range is from -10 to 10.
Yes. If the discriminant equals zero, the calculator identifies repeated roots and clearly labels the repeated value.
Yes. Use the “Clear all” button to reset inputs and hide results, or use the “Reload calculator” button to fully restore the default state.
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