How to Solve Cubic Equations Online with Step-by-Step Solutions
Cubic Equation Solver
What is Cubic Equation Solver?
A Cubic Equation Solver is a fast online tool that finds the solutions (roots) for any cubic equation of the form ax³ + bx² + cx + d = 0. Unlike a simple calculator, this solver instantly calculates all real and complex roots and shows each solving step using advanced algebraic methods like Cardano’s formula. Whether you’re working on academics, competitive exams, or engineering assignments, this interactive tool helps you conquer third degree equations efficiently.
Formula or Logic Behind Cubic Equation Solver
The general cubic equation is written as ax³ + bx² + cx + d = 0 (with a ≠ 0). Solving it analytically uses the Cardano’s method:
1. Change variables to transform into a depressed cubic (remove x² term): Let x = t - (b/3a).
2. The depressed cubic becomes: t³ + pt + q = 0 where:
p = (3ac - b²)/(3a²),
q = (2b³ - 9abc + 27a²d)/(27a³)
3. The roots are then found using:
t = u + v, where u = ∛[ -q/2 + √(q²/4 + p³/27) ], v = ∛[ -q/2 - √(q²/4 + p³/27) ]
4. Substitute back to get x.
Depending on the cubic discriminant, the equation can have one real & two complex roots, or all three real roots (possibly repeated).
Sum & Product of Roots:
If r₁, r₂, r₃ are roots:
r₁ + r₂ + r₃ = -b/a
r₁·r₂ + r₂·r₃ + r₁·r₃ = c/a
r₁·r₂·r₃ = -d/a
Solved Examples of Cubic Equations
| Coefficients (a,b,c,d) | Equation | Roots | Steps (Short) |
|---|---|---|---|
| 1, -6, 11, -6 | x³ - 6x² + 11x - 6 = 0 | 1, 2, 3 | Factorizable: (x-1)(x-2)(x-3)=0 |
| 2, -4, 3, -5 | 2x³ - 4x² + 3x - 5 = 0 | One real, two complex (roots shown by tool) |
Cardano’s formula |
| 1, 0, 0, -8 | x³ - 8 = 0 | 2, -1 + i√3, -1 - i√3 | Roots: ∛8, ∛8·ω, ∛8·ω²; where ω=complex cube root |
| 1, -3, 3, -1 | x³ - 3x² + 3x - 1 = 0 | 1 (triple root) | (x-1)³ = 0 |
Steps to Use the Cubic Equation Solver
- Enter values for the coefficients a, b, c, and d in the respective fields above.
- Click on the 'Calculate' button.
- Get instant results for all real and complex roots, with step-by-step explanation.
Why Use Vedantu’s Cubic Equation Solver?
Vedantu’s Cubic Equation Solver is designed for maximum clarity and lightning-fast calculations. It’s mobile-friendly, fully responsive, and provides not only the final roots but also explains each solving step so students can learn as they compute. Trusted by lakhs of students for JEE, CBSE, Olympiads, and daily homework, this tool simplifies even the toughest third degree equations and works right inside your browser, with no signup required.
Real-life Applications of Cubic Equation Solver
Cubic equations often appear in real-world math problems, including physics (projectile and trajectory calculations), engineering (designing curves and gears), economics (modeling profit/cost functions), chemistry (reaction rates), and computer graphics (Bezier curves, animations). Quick and accurate solving saves time in exams, competitive tests, and STEM research or projects.
You’ll also use cubic equations in:
- Math Olympiad and IIT JEE entrance preparation
- Solving real-life business or finance equations
- Computer programming and algorithm analysis
- Science laboratory data fitting or curve modeling
- Higher studies in algebra, calculus, and engineering
Want more help with algebra? Try our related resources: Quadratic Equation Solver, Prime Numbers, Factorization Methods, and explore Vedantu's Algebra Topics for complete learning support.
FAQs on Cubic Equation Solver: Free Online Calculator & Stepwise Solution
1. How does an online cubic equation solver work?
An online cubic equation solver simplifies the process of finding roots for a cubic equation. You need to input the coefficients (the numerical values of a, b, c, and d) from your equation into the calculator. The tool then applies sophisticated mathematical algorithms, such as Cardano's method or numerical approximations, to compute the three roots. Many solvers also provide a step-by-step breakdown of the solution process.
2. What is a cubic equation, and what are its roots?
A cubic equation is a third-degree polynomial equation of the standard form ax³ + bx² + cx + d = 0, where 'a' is not equal to zero. The roots of the equation (also known as 'solutions' or 'zeroes') are the specific values of 'x' that satisfy the equation. Every cubic equation has exactly three roots, which can be a combination of real and complex numbers.
3. What is the general formula for a cubic equation?
The general formula, or standard form, for a cubic equation is ax³ + bx² + cx + d = 0. In this formula:
- x is the variable.
- a, b, and c are the coefficients of the terms x³, x², and x, respectively.
- d is the constant term.
- The coefficient 'a' cannot be zero, otherwise the equation would become quadratic.
4. How do I use the calculator to find the roots of a cubic equation?
To find the roots using an online calculator, follow these simple steps:
- First, write your equation in the standard form ax³ + bx² + cx + d = 0 to correctly identify the coefficients.
- Enter the values for a, b, c, and d into the designated input fields on the calculator.
- Click the 'Solve' or 'Calculate' button.
- The tool will instantly display the three roots of your cubic equation.
5. Can you provide an example of solving a cubic equation step-by-step?
Certainly. Let's take the equation x³ - 2x² - 5x + 6 = 0. Here, the coefficients are a=1, b=-2, c=-5, and d=6. An online calculator would find the roots using methods like the Rational Root Theorem. It would identify that x=1 is a root. By dividing the polynomial by (x-1), it reduces to the quadratic equation x² - x - 6 = 0. Solving this quadratic equation gives the remaining roots, x=3 and x=-2. Thus, the three roots are x = 1, x = 3, and x = -2.
6. Why can't all cubic equations be solved by simple factoring like quadratic equations?
While some simple cubic equations can be solved by grouping or using the Factor Theorem, most cannot. The relationship between the coefficients and the three roots in a cubic equation is far more complex than in a quadratic equation. There isn't a straightforward method like 'splitting the middle term' that works universally. This complexity is why advanced algebraic methods, like Cardano's method, or numerical calculators are essential for finding the roots of most cubic equations.
7. Can a cubic equation have three imaginary roots? Why or why not?
No, a cubic equation with real coefficients cannot have three imaginary (complex) roots. This is because complex roots of polynomials with real coefficients always occur in complex conjugate pairs. Therefore, a cubic equation can only have two possible combinations of roots:
- Three real roots.
- One real root and a pair of two complex conjugate roots.
8. How is solving a cubic equation related to factoring it?
Solving and factoring a cubic equation are two sides of the same coin, linked by the Factor Theorem. This theorem states that if 'r' is a root of the equation, then (x - r) must be a factor of the polynomial. When a solver finds the three roots (let's call them r₁, r₂, and r₃), it has essentially found the components needed to write the polynomial in its factored form: a(x - r₁)(x - r₂)(x - r₃). So, finding the roots is the key to factoring the equation.
9. What advanced method does a cubic equation calculator use to find the roots?
For an exact algebraic solution, an advanced cubic equation calculator typically automates a process called Cardano's method. This is the 'cubic formula' equivalent to the quadratic formula, but it is extremely long and involves complex number calculations even when the final roots are real. For equations where this is too cumbersome or for providing simpler step-by-step solutions, the calculator may first apply the Rational Root Theorem to find any integer/rational roots before solving the remaining part.
10. Is there a 'cubic formula' that works like the quadratic formula?
Yes, a direct formula for solving any cubic equation exists, and it is most commonly known as Cardano's method. However, it is vastly more complicated than the well-known quadratic formula. Its manual application is prone to errors and involves intermediate steps with complex numbers. The existence of this complex formula is precisely why a cubic equation solver is such a valuable and time-saving tool for students and professionals, as it handles all the intricate calculations automatically.
