Find Three Solutions Of The Equation Calculator

Enter the coefficients of your cubic equation ax³ + bx² + cx + d = 0 to find its real roots using our Cubic Equation Solver.

Cubic Equation Solver

Coefficients and Intermediates Table

Item	Value
a	1
b	-6
c	11
d	-6
p	N/A
q	N/A
Δ	N/A

Table showing the input coefficients and calculated intermediate values.

Cubic Function Graph

Graph of f(x) = ax³ + bx² + cx + d around the real roots. Red dots indicate real roots found by the Cubic Equation Solver.

What is a Cubic Equation Solver?

A Cubic Equation Solver is a tool designed to find the roots (or solutions) of a cubic equation, which is a polynomial equation of the third degree. The general form of a cubic equation is ax³ + bx² + cx + d = 0, where 'a', 'b', 'c', and 'd' are coefficients, and 'a' is non-zero. A Cubic Equation Solver helps determine the values of 'x' that satisfy this equation.

This type of solver is used by students, engineers, scientists, and mathematicians who encounter cubic equations in various fields like physics, engineering, economics, and computer graphics. The Cubic Equation Solver automates the process of finding up to three real roots or one real root and two complex roots.

Common misconceptions include thinking that all cubic equations have three distinct real roots; some have one real root and two complex roots, or repeated real roots. Our Cubic Equation Solver focuses on finding the real roots.

Cubic Equation Solver Formula and Mathematical Explanation

To solve the cubic equation ax³ + bx² + cx + d = 0 (where a ≠ 0), we first simplify it:

1. Divide by 'a': x³ + (b/a)x² + (c/a)x + (d/a) = 0. Let A = b/a, B = c/a, C = d/a, so x³ + Ax² + Bx + C = 0.
2. Substitute x = y – A/3 to eliminate the x² term, resulting in the depressed cubic equation: y³ + py + q = 0, where:
   • p = B – A²/3 = c/a – (b/a)²/3
   • q = C + (2A³ – 9AB)/27 = d/a + (2(b/a)³ – 9(b/a)(c/a))/27
3. Calculate the discriminant Δ = q²/4 + p³/27. The sign of Δ determines the nature of the roots:
   • If Δ > 0, there is one real root and two complex conjugate roots.
   • If Δ = 0, there are three real roots, with at least two being equal.
   • If Δ < 0, there are three distinct real roots (the "casus irreducibilis", requiring trigonometric solution for real roots).

Finding the real roots:

• If Δ ≥ 0: One real root can be found using Cardano's method, and if Δ=0, the other roots are also real and easily derived.
• If Δ < 0: All three roots are real and distinct, found using Vieta's trigonometric solution: y_k = 2 * sqrt(-p/3) * cos((1/3) * arccos(-q / (2 * sqrt(-p³/27))) + 2πk/3) for k = 0, 1, 2. Then x_k = y_k – A/3.

Variables Table

Variable	Meaning	Unit	Typical range
a	Coefficient of x³	None	Any non-zero real number
b	Coefficient of x²	None	Any real number
c	Coefficient of x	None	Any real number
d	Constant term	None	Any real number
p, q	Coefficients of the depressed cubic	None	Real numbers
Δ	Discriminant	None	Real number
x1, x2, x3	Real roots of the equation	None	Real numbers

Practical Examples (Real-World Use Cases)

While cubic equations might seem abstract, they appear in various real-world scenarios.

Example 1: Finding Equilibrium Points

In certain physical or economic models, equilibrium points are found by solving polynomial equations. Suppose a system's state is described by x³ – 6x² + 11x – 6 = 0. Using the Cubic Equation Solver with a=1, b=-6, c=11, d=-6, we find the real roots x1=1, x2=2, and x3=3, representing stable or unstable equilibrium points.

Example 2: Volume and Dimensions

Imagine a box with volume given by V(x) = x(x+1)(x-4) = x³ – 3x² – 4x. If the volume is -6, we solve x³ – 3x² – 4x + 6 = 0. Let's say we have x³ – 7x² + 14x – 8 = 0. Using the Cubic Equation Solver with a=1, b=-7, c=14, d=-8, we get roots x1=1, x2=2, x3=4, which could relate to possible dimensions or parameters.

How to Use This Cubic Equation Solver

1. Enter Coefficients: Input the values for 'a', 'b', 'c', and 'd' from your equation ax³ + bx² + cx + d = 0 into the respective fields. 'a' cannot be zero.
2. Real-time Calculation: The calculator automatically updates the solutions and intermediate values (p, q, Δ) as you type.
3. View Results: The "Results" section will show the number of real roots found and their values (x1, x2, x3). If there's only one real root, x2 and x3 will be indicated as not real or not applicable in this real-root solver context for Δ > 0.
4. Check Intermediates: The values of 'p', 'q', and the discriminant 'Δ' are displayed to give insight into the solution method.
5. See the Graph: The graph plots the function f(x) = ax³ + bx² + cx + d, visually indicating where the function crosses the x-axis (the real roots).
6. Reset: Use the "Reset" button to clear the inputs to their default values.
7. Copy Results: Use the "Copy Results" button to copy the coefficients, roots, and intermediate values to your clipboard.

Understanding the results from the Cubic Equation Solver helps in analyzing the behavior of the system described by the equation.

Key Factors That Affect Cubic Equation Solver Results

• Coefficient 'a': Although it can be scaled, its sign affects the general shape of the cubic function (rising or falling at large |x|). It cannot be zero for a cubic equation.
• Coefficient 'b': Influences the position of the inflection point and the x² term's contribution.
• Coefficient 'c': Affects the slope at various points and the linear contribution.
• Coefficient 'd': The y-intercept; it shifts the graph vertically, directly impacting the roots' values.
• Relative Magnitudes of Coefficients: The interplay between a, b, c, and d determines the values of p, q, and Δ, and thus the nature and location of the roots.
• The Discriminant (Δ): The sign of Δ = q²/4 + p³/27 is crucial. Δ > 0 means one real root; Δ = 0 means three real roots (at least two equal); Δ < 0 means three distinct real roots. Our Cubic Equation Solver is designed to find these real roots.

Frequently Asked Questions (FAQ)

What is a cubic equation?

A cubic equation is a polynomial equation of the third degree, meaning the highest power of the variable is 3. Its general form is ax³ + bx² + cx + d = 0, where a ≠ 0.

How many solutions does a cubic equation have?

A cubic equation always has three roots (solutions) in the complex number system. However, it can have one, two (one distinct, one repeated), or three distinct real roots. Our Cubic Equation Solver focuses on finding the real roots.

What if 'a' is zero?

If 'a' is zero, the equation becomes bx² + cx + d = 0, which is a quadratic equation, not cubic. You would need a {related_keywords[0]} for that.

What does the discriminant (Δ) tell us?

The discriminant (Δ = q²/4 + p³/27) of the depressed cubic tells us about the nature of the roots: Δ > 0 (one real root, two complex), Δ = 0 (three real roots, at least two equal), Δ < 0 (three distinct real roots).

Can this Cubic Equation Solver find complex roots?

This particular Cubic Equation Solver is focused on finding and displaying the real roots. When Δ > 0, there are complex roots which are not explicitly calculated here, but their existence is implied.

What is the 'depressed cubic'?

The depressed cubic is a simplified form y³ + py + q = 0, obtained from the original cubic by a variable substitution to eliminate the x² term, making it easier to solve.

Why use a Cubic Equation Solver?

Solving cubic equations by hand can be complex and time-consuming, especially for the case with three real roots (casus irreducibilis). A Cubic Equation Solver provides quick and accurate solutions.

Are there limitations to this calculator?

This calculator finds real roots and is subject to the precision limits of standard floating-point arithmetic. For very large or very small coefficients, precision issues might arise.

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