Quartic Equation Calculator

Quartic Equation Calculator

Quartic equation calculator evaluates the equations that include a variable having “4” in its exponent, it could be x^4 or ax^4 + 1, etc.

What is Quartic Equation?

A quartic equation is a polynomial equation of degree four. It is expressed as:

ax⁴ + bx³ + cx² + dx + e = 0

Here, the coefficients (a, b, c, d, e) can be real numbers or complex numbers. The variable x represents the unknown quantity. Quartic equations can have multiple solutions, including real and complex roots.

General Form of a Quartic Equation:

The general form of a quartic equation is given by:

x⁴ + ax³ + bx² + cx + d = 0

This form helps in understanding the structure of a quartic equation and its various coefficients.

What is Polynomial Equation?

Before diving into quartic equations, it's important to have a basic understanding of polynomial equations. Polynomial equations consist of algebraic expressions with multiple terms involving variables raised to non-negative integer powers.

The highest power of the variable in a polynomial equation determines its degree. Quartic equations, specifically, are fourth-degree polynomial equations.

Solving Quartic Equations:

Solving quartic equations can be a challenging task due to their higher degree. Several methods have been developed over the years to find the solutions to these equations. Let's explore some of the popular methods used for solving quartic equations.

Factorization Method:

One of the simplest methods to solve quartic equations is through factorization. In this approach, the quartic equation is factored into two quadratic equations, which are then solved individually. The solutions obtained from the quadratic equations are the roots of the quartic equation.

The Depressed Quartic Method:

The depressed quartic method involves transforming the original quartic equation into a depressed quartic equation, which has a simplified form. This method simplifies the process of solving quartic equations by reducing the degree of the polynomial.

The depressed quartic equation is then solved using various techniques, such as substitution or trigonometric methods.

Numerical Methods: Newton's Method and Iterative Methods

When analytical methods fail to provide exact solutions, numerical methods come into play. Newton's method, a popular numerical technique, involves iteratively refining an initial guess to approach the roots of the quartic equation.

Iterative methods, such as the bisection method or the secant method, also provide approximate solutions to quartic equations.

How to solve Quartic Equations?

Example 

Solve the quartic equation:

2x⁴ - 5x³ + 3x² + 2x - 1 = 0

Solution:

Specify the method

To solve this equation, we can use the factorization method. Let's factorize it into two quadratic equations:

(2x² + x - 1) (x² - 2x + 1) = 0

Solving the first quadratic equation, we have:

2x² + x - 1 = 0

Using the quadratic formula, the solutions are:

x = (-1 ± √ (1 - 4*2*(-1))) / (2*2)

x = (-1 ± √ (9)) / 4

x = (-1 ± 3) / 4

So, the solutions for the first quadratic equation are:

x = 1/2 or x = -1

Now, let's solve the second quadratic equation:

x² - 2x + 1 = 0

This equation can be factored as:

(x - 1) (x - 1) = 0

So, the solution for the second quadratic equation is:

x = 1

Therefore, the solutions to the original quartic equation are:

x = 1/2, -1, 1

These are the values of x that satisfy the quartic equation.