Cubic Simultaneous Equation Solver With Steps
Numerical estimates use finite precision. Root scans, sign charts and calculus can miss narrow features; verify sensitive results independently. Angles use radians; log is base 10 and ln is natural log.
CALCULATOR OVERVIEW
Solve Cubic Equations and Linear Systems
Use the cubic equation solver provided above to find the real and complex roots of a cubic equation. For systems of linear equations, switch to simultaneous mode and select two, three, or four variables.
Next, enter the numerical coefficients, choose a solution method, and view the result along with a summary of the procedure.
Once solved, you can copy the equation or download a text report. However, JavaScript must be enabled in your browser to perform the calculations.
For a system comprising multiple equations, the Solver tool in an online spreadsheet editor can assist you in arranging the coefficients in relation to the variables.
CHOOSE YOUR SOLVER
Choose the mode that matches your equation
Start with the structure of your problem. A cubic equation involves an unknown raised to the third power. In contrast, systems of simultaneous linear equations consist of multiple equations that must apply to the same set of variable values.
Cubic: Enter a, b, c, and d representing ax³ + bx² + cx + d = 0, where a ≠ 0.
Simultaneous: Enter a linear equation for each row using the same order of variables.
Quadratic: Use this mode if the highest power is two.
Polynomials: For other problems related to the roots of polynomials, select a power between 2 and 10.
Note: The cubic and simultaneous tabs perform distinct functions. The simultaneous mode cannot solve nonlinear systems that include terms such as x², xy, or x³.
COEFFICIENT INPUT
Enter a cubic equation in the correct form
Move every term to the left side so that the right side becomes zero. Then, write down the coefficient associated with each power of x, including its sign. In other words, enter only the coefficients in these boxes, rather than the entire equation.
If 2x³ − 5x = 7, then first write 2x³ + 0x² − 5x − 7 = 0. Enter:
Field
Term
Value
a
x³
2
b
x²
0
c
x
−5
d
Constant
−7
Once all four fields are filled, select Solve. If a term is missing, enter zero (0) because the field cannot be left blank. However, when a = 0, the equation is no longer cubic (third-degree). Therefore, select the appropriate mode for a lower-degree equation.
Tip: Convert coefficients given as fractions into decimal form before entering them. For example, enter 0.5 for 1/2. Ensure that the number of decimal places used does not alter the nature of the intended equation.
CUBIC WORKED EXAMPLE
Check a cubic with three real roots
If x³ − 6x² + 11x − 6 = 0, then set a = 1, b = −6, c = 11, and d = −6. Its roots are x = 1, x = 2, and x = 3. The solution does not change based on the order in which they appear.
Manual inspection makes it clear why these values are useful. The factors of the polynomial are (x − 1)(x − 2)(x − 3). Therefore, each value being substituted makes one of these factors zero.
At x = 1: 1 − 6 + 11 − 6 = 0.
For x = 2: 8 - 24 + 22 - 6 = 0.
At x = 3: 27 - 54 + 33 - 6 = 0.
This result also offers an opportunity to verify the coefficients: the sum of the roots is 6, the sum of the products of the roots taken in pairs is 11, and their product is 6. For example, this factorization shows how to verify the roots manually. However, this tool uses its own specific method to solve the cubic equation, rather than displaying the factorization process itself.
UNDERSTAND THE ROOTS
Understand the different types of cubic roots
When counted with multiplicity, a cubic equation with real coefficients has three roots. They may all be real or consist of one real root and a pair of complex roots. A repeated root is a value that appears more than once.
Real root: The value that makes the cube equal to zero, for example, x = 2.
Complex pair: values of the form u + vi and u − vi, where i² = −1.
Repeated root: For the equation x³ − 3x² + 3x − 1 = 0, x = 1 occurs three times because the expression is (x − 1)³.
In the Show Step-by-Step view, this tool displays the discriminant D = (q/2)² + (p/3)³ of the depressed cubic.
The cases are as follows: D > 0 indicates one real root and a pair of complex conjugate roots, D < 0 reflects three distinct real roots, and D = 0 signifies repeated roots.
For the basic concept, refer to the section of OpenStax that deals with the zeros of polynomial functions.
Note: Some textbooks use different notations for the discriminant and for opposite signs. Compare the formulas and notations.
SIMULTANEOUS LINEAR EQUATIONS
Solve a linear system and verify both equations
Select Simultaneous, then choose one of the options 2 (x, y), 3 (x, y, z), or 4 (x, y, z, w). Each row represents an equation. After that, enter the coefficients of the variables and the constants on the right-hand side into the respective columns.
For x + y = 5 and 2x − y = 1, choose two variables and enter them.
Row
x coefficient
y coefficient
Constant
Equation 1
1
1
5
Equation 2
2
−1
1
The solution is x = 2 and y = 3. Adding the equations manually yields 3x = 6, so x = 2. Substituting this value of x into x + y = 5 yields y = 3.
First equation: 2 + 3 = 5.
Second equation: 2(2) − 3 = 1.
This solver uses the Gaussian elimination method for linear systems. Its expandable steps summarize the procedure, while the manual calculation provided above serves as a separate example for learning purposes.
SYSTEM SOLUTION TYPES
Distinguish one solution from no unique solution
A linear system can have a unique solution, no solution, or infinitely many solutions. In the latter two cases, there is no single specific answer, yet their implications differ. To illustrate these outcomes, consider the following examples involving two variables:
One solution: The pair (2, 3) is obtained from the equations x + y = 5 and 2x − y = 1. This pair satisfies both equations.
No solution: x + y = 5 and 2x + 2y = 12 contradict each other. Doubling the first equation gives 2x + 2y = 10 instead of 12.
Infinitely many solutions: x + y = 5 and 2x + 2y = 10 represent the same condition. For any real number t, the pair (t, 5 − t) satisfies both equations.
The Infinite-solutions message neither selects a single preferred pair nor provides a list of all possible answers.
In contrast, the no-solution message indicates that no combination of values satisfies the entire system. Reverify the entered data before using the report to determine the expected result.
MORE MATH MODES
Use the other solvers for related calculations
Open More Solvers when a problem requires specific inputs or outputs. Instead of trying to fit every problem into cubic or simultaneous modes, select the mode best suited for the specific calculation.
Finding the roots of quadratic and polynomial equations includes support for polynomials with degrees ranging from 2 to 10 and non-simple symbolic expressions.
This matrix processes 2×2 and 3×3 inputs to calculate the determinant, trace, eigenvalues, rank, and the reciprocal of A × B.
Statistics provides a summary of a collection of numerical data. Number theory includes topics such as GCD, LCM, primality, and factorization.
For additional statistical analysis—including correlations, linear regression, and graphs—consider SPSS Lite.
These expressions determine the value of f(x) or identify real roots, critical points, and sign information for a selected interval.
Calculus examines derivatives and definite integrals, including extrema and tangents. Inequalities return the solution intervals for linear or quadratic inequalities.
Note: Numerical methods are used to find the roots of an expression, and some roots may be missed. The results from calculus are numerical approximations, not the outcome of symbolic differentiation or integration.
Check out the NexezTool Suite for relevant spreadsheets, data, and calculation tools.
REVIEW AND SAVE
Review the method and keep a result report
After selecting the solution, view the roots or variable values and open the step-by-step view for a summary of the procedure. Cubic mode displays the depressed cubic and its discriminant, while Simultaneous mode summarizes Gaussian elimination and the system's results.
Copy Equation creates a copy of the current equation representation so that you can preserve the problem statement.
The Download Result option saves a .txt report upon successful completion. This is a text file, not a PDF.
If you want to test the input state, use the example to load the selected state.
Finally, reset the workspace to prepare it for another calculation.
For homework exercises, compare your work with the final answer and substitute the results into the original problem. The detailed explanation provides an algorithmic overview. It does not offer a step-by-step account of manual factorization or row operations performed on matrices.
Suggestion: After changing the input, select Resolve again before downloading. This report relates to the calculation you have just resolved.
INPUT CHECKS AND PRECISION
Check the input when a result looks unexpected
First, compare the recorded values with the original equation. A sign change, a missing zero, or the misplacement of a constant can lead to further problems, even if the result appears correct at first glance.
Fill in all the required coefficient fields with the final values and retain the negative signs.
Also, maintain a consistent order of variables in every equation of the system. If the term containing y is missing, it is necessary to use 0 as the coefficient.
Carefully move this term to the other side of the equals (=) sign. For example, the equation y = 2x + 1 becomes −2x + y = 1.
Use decimal factors instead of fractions (such as 1/3) and avoid rounding intermediate values prematurely.
Computer arithmetic has finite precision. Nearly repeated roots, nearly dependent equations, or coefficients of vastly different magnitudes can make the calculation sensitive to rounding errors. If the solver reports validation or convergence issues, check the equations, scale them consistently, or look for a simpler factorization.
Note: When an equation is divided by a non-zero constant, the solution remains unchanged only if all terms are divided, including the constant on the right-hand side in linear systems.
QUICK ANSWERS
Frequently asked questions
Can the entire equation be placed within the coefficient field?
There is no need for that. Separate slots are used for cubic equations and simultaneous equations. There is a dedicated $f(x)$ slot for expressions like $x^2 - 5x + 6$. It is simply a different way of working.
Why can there be an i in a cube root?
This equation can have unrealistic solutions. For example, the roots of x³ − 1 = 0 are 1 and −1/2 ± (√3/2)i. These pairs are part of the solution, not a typing error.
Are roots and turning points the same thing?
No. The roots correspond to f(x) = 0. Stationary points correspond to f′(x) = 0, and some of these are turning points. Use appropriate formulas or computational methods when conducting your analysis.
Where are accounts settled?
The solver performs calculations within your browser. Meanwhile, the website’s cookies, analytics, and advertisements operate separately from the solver’s calculations. Please refer to the website's privacy policy for information regarding these matters.
What will happen if the calculator does not show any response?
Ensure that JavaScript is enabled, fill in the required fields, and try out the example in the selected mode. If that does not work, reload the page and check if any browser extension is blocking the script.
