Quartic equation

Introduction

A quartic equation is a polynomial equation of degree four. It takes the general form: \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] where \( a, b, c, d, \) and \( e \) are constants, and \( a \neq 0 \). Quartic equations are also known as biquadratic equations or fourth-degree equations. Solving these equations involves finding the values of \( x \) that satisfy the equation.

Historical Background

The study of quartic equations dates back to ancient civilizations, but significant progress was made during the Renaissance. The first general solution to the quartic equation was discovered by the Italian mathematician Lodovico Ferrari in 1540. Ferrari's work built upon the earlier solutions to cubic equations by his mentor, Tartaglia, and Gerolamo Cardano.

General Form and Properties

A quartic equation can be expressed in its general form: \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] where \( a \neq 0 \). The properties of quartic equations include:

**Degree**: The highest power of the variable \( x \) is four.
**Roots**: A quartic equation can have up to four real roots or a combination of real and complex roots.
**Symmetry**: Quartic equations can exhibit symmetry, particularly if they are biquadratic (i.e., they lack the \( x^3 \) and \( x \) terms).

Solving Quartic Equations

Solving quartic equations can be approached through various methods, including:

Ferrari's Method

Ferrari's method involves reducing the quartic equation to a cubic equation, which can then be solved using known techniques. The steps are as follows:

1. **Depress the Quartic**: Transform the quartic equation into a depressed quartic equation (one without the \( x^3 \) term) by substituting \( x = y - \frac{b}{4a} \). 2. **Solve the Depressed Quartic**: Use Ferrari's method to solve the depressed quartic equation.

Factoring

In some cases, quartic equations can be factored into products of lower-degree polynomials. For example: \[ ax^4 + bx^3 + cx^2 + dx + e = (px^2 + qx + r)(sx^2 + tx + u) \]

Numerical Methods

When analytical solutions are difficult to obtain, numerical methods such as Newton's method or Bairstow's method can be employed to approximate the roots.

Special Cases

Certain forms of quartic equations allow for simpler solutions:

Biquadratic Equations

A biquadratic equation has the form: \[ ax^4 + cx^2 + e = 0 \] These can be solved by substituting \( z = x^2 \), reducing the equation to a quadratic in \( z \).