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Quartic Equation Calculator

What Is a Quartic Equation?

A quartic equation is a polynomial equation of degree 4. This means it includes a variable raised to the fourth power, such as x⁴. The general form of a quartic equation is:

ax⁴ + bx³ + cx² + dx + e = 0

In this equation:

a is the coefficient of x⁴ and must not be zero, or the equation wouldn't be quartic.
b is the coefficient of x³.
c is the coefficient of x².
d is the coefficient of x.
e is the constant term (also called the independent term).

A quartic equation can have up to four solutions (also called roots), which may be:

All real numbers
A mix of real and complex numbers
All complex numbers (including imaginary parts)

Quartic equations are commonly found in advanced math, physics, engineering, and certain real-life applications, such as structural design, optics, and finance. Unlike linear, quadratic, or even cubic equations, quartic equations are more challenging to solve manually because they require advanced algebraic techniques.

Sometimes, quartic equations can be simplified if they have a special form, like a biquadratic equation (no x³ or x terms), but in general, solving them involves several steps — such as reducing the equation to a simpler form, solving a related cubic equation, and handling complex numbers if necessary.

The Quartic Equation Calculator provided on this page is designed to make that process easy. You simply enter the values of the coefficients a, b, c, d, and e, and the calculator will do the heavy lifting — providing you with all four roots, whether they’re real or complex.

Purpose of the Calculator

The Quartic Equation Calculator is designed to help you solve fourth-degree polynomial equations quickly and accurately, without the need for complicated algebra or lengthy manual calculations.

Given an equation in the form:

ax⁴ + bx³ + cx² + dx + e = 0

you can simply enter the values of the coefficients a, b, c, d, and e into the calculator. Once you click the "Solve" button, the tool will instantly compute all four solutions (also known as roots) of the equation.

These roots may be real numbers or complex numbers (with imaginary parts). The calculator will display each root clearly, separating the real and imaginary components so you can easily understand the result.

This tool is especially useful for:

Students learning algebra and higher-level mathematics
Teachers and tutors who need to check homework or explain quartic equations
Professionals and engineers solving technical problems involving fourth-degree equations
Anyone curious about the behavior of polynomial functions

Input Fields Explained

To use the Quartic Equation Calculator, you need to enter five numerical values that make up the coefficients of your equation. These inputs represent the different parts of the quartic equation:

Coefficient a (x⁴ term): This is the leading coefficient, and it must not be zero. It determines the shape and width of the curve. Example: 1 or -3.5
Coefficient b (x³ term): This value affects the symmetry and turning points of the graph. Example: 0, 2, or -7
Coefficient c (x² term): This controls how steep the curve rises or falls in the middle of the equation. Example: 1, -4.25
Coefficient d (x term): This influences the slope and tilt of the curve. You can enter any real number, including 0. Example: -2, 3.75
Constant e: This is the final term in the equation with no variable. It shifts the curve up or down. Example: 0, 6.1, or -1

Input Tips:

Use standard numbers only — no fractions or symbols (e.g., enter 0.5 instead of ½).
All fields accept decimal numbers, both positive and negative.
If a certain term is missing in your equation (e.g., no x³ term), just enter 0 in that field.
Make sure the first coefficient (a) is not zero — otherwise, the equation will no longer be quartic.

How to Use the Calculator

Using the Quartic Equation Calculator is simple and requires only three easy steps. Follow the instructions below to find the roots of any fourth-degree polynomial equation:

Enter Your Coefficients:
In the input fields provided, type the values for the coefficients a, b, c, d, and e. These correspond to the equation:

ax⁴ + bx³ + cx² + dx + e = 0
If a term is missing in your equation, simply enter 0 for that coefficient.
Click the “Solve” Button:
Once all fields are filled, press the Solve button. The calculator will instantly process your input and apply a reliable algorithm to find the equation’s four solutions.
View Your Results:
The four roots will appear below the form. Each solution will be displayed clearly, including both the real and imaginary parts (if applicable). Complex roots will include an i to indicate the imaginary unit.

That’s it! No need for manual calculations, formulas, or guesswork — the calculator handles it all for you with just a few clicks.

How the Calculation Works (Simplified)

Behind the scenes, the Quartic Equation Calculator follows a multi-step mathematical process to accurately find the roots of your equation. While the full method involves advanced algebra, here’s a simplified explanation of what’s happening:

1. Normalizing the Coefficients

To make the equation easier to handle, the calculator first divides all coefficients by the leading coefficient a (as long as it's not 1). This transforms the equation so that the x⁴ term has a coefficient of 1. This step helps simplify the next phases of the calculation.

2. Reducing to a Depressed Quartic

Next, the calculator rewrites the equation in a special form called a depressed quartic. This form eliminates the x³ term, making it easier to solve. It uses a change of variable to rewrite the equation in a cleaner format that focuses on the remaining terms.

3. Solving an Intermediate Cubic Equation

To solve the depressed quartic, the calculator needs to find a helpful value known as a resolvent cubic. This is a cubic (third-degree) equation that is derived from the quartic equation. Solving this intermediate cubic allows the tool to move closer to the actual roots of the original quartic equation.

4. Extracting Real and Complex Roots

Once the cubic equation is solved, the calculator uses those results to compute the four roots of the original equation. These roots may be:

Real numbers (standard values on the number line)
Complex numbers (values that include an imaginary part, such as 3 + 2i)

The calculator carefully handles both possibilities and displays each root clearly, separating the real and imaginary parts for your convenience.

All of these steps happen instantly when you click the "Solve" button — no manual math required!

Understanding the Output

After you click the "Solve" button, the calculator displays the four roots of your quartic equation. These roots are shown as:

x₁, x₂, x₃, x₄

Each root is separated into two parts:

Real part: The actual value on the number line.
Imaginary part (if any): A value followed by the letter i, which represents the square root of -1.

For example, a solution might look like this:

x₁: 2.45 + 0.00i
x₂: -1.22 + 0.00i
x₃: 0.38 + 1.97i
x₄: 0.38 - 1.97i

In this case:

x₁ and x₂ are real roots (imaginary part is 0).
x₃ and x₄ are complex roots (they include both real and imaginary parts).

When Do Complex Roots Appear?

Complex roots appear when the equation does not cross the x-axis at enough points to give four real roots. This happens when the quartic curve lies entirely above or below the x-axis in some sections.

Also, complex roots for real-coefficient equations always come in conjugate pairs. That means if one root is a + bi, the other will be a - bi.

If all four roots are real, the calculator will show all imaginary parts as 0.00i. If two or more roots are complex, you will see non-zero imaginary parts for those values.

This clear breakdown helps you understand the complete behavior of your quartic equation — whether the solutions are purely real, purely imaginary, or a mix of both.

Special Cases Handled Automatically

The Quartic Equation Calculator is smart enough to detect and solve several special types of equations without any extra steps from you. These include:

1. Perfect Quartic Solutions

Sometimes, a quartic equation has four identical roots — for example:

(x + 2)⁴ = 0

This expands to a quartic equation with all four roots being the same value: -2. The calculator recognizes this pattern and shows all four roots with the same result. It also ensures that the imaginary parts are shown as 0.00i, since the solutions are purely real.

2. Biquadratic Forms

A biquadratic equation is a special type of quartic equation that only includes even powers of x, such as:

ax⁴ + cx² + e = 0

There are no x³ or x terms. These equations are often easier to solve by treating x² as a variable. The calculator detects this form and simplifies the process to provide accurate solutions, even if they include complex numbers.

3. Multiple or Repeated Roots

In some equations, two or more roots may be the same. For example, the equation:

(x - 1)² (x + 2)² = 0

has repeated roots: x = 1 (twice) and x = -2 (twice). The calculator automatically detects these repeated roots and shows them clearly in the output. Each root will still be listed individually, even if some are duplicates.

These built-in features make the calculator powerful and flexible, so you don’t need to worry about identifying or simplifying special cases manually — it’s all handled for you!

Troubleshooting and Tips

If you're getting unexpected results or confused by the output, don’t worry — here are some tips and common issues to check:

What to Do If You See "NaN" or Strange Results

NaN stands for "Not a Number." It usually appears when the calculator encounters an invalid math operation, such as dividing by zero or trying to take the square root of a negative number incorrectly.
Double-check all input fields. Make sure you’ve entered only valid numbers and that the coefficient a (for x⁴) is not zero.
Try refreshing the page and re-entering the values from scratch in case of any glitches.

Why Some Roots May Be Complex

Not all quartic equations have only real solutions. Depending on the values of the coefficients, some roots may involve imaginary numbers (e.g., 3 + 2i).
This is normal and expected behavior in advanced mathematics. If the equation’s curve doesn’t intersect the x-axis in all places, complex roots can occur.
The calculator handles this automatically and clearly displays real and imaginary parts separately so you can understand the result easily.

Common Input Errors to Avoid

Leaving a required field blank — always enter a number in each coefficient box, even if it’s 0.
Entering non-numeric characters (like letters or symbols) — the calculator only accepts numbers, including decimals and negative values.
Setting a = 0 — this turns the equation into a cubic, not quartic. Make sure the coefficient for x⁴ is not zero.
Using commas instead of dots for decimals (use 1.5, not 1,5).

If you're still unsure about the result, try changing your input slightly to test different values or consult an example to compare.