Biquadratic Substitutions

Biquadratic equations and their substitutions are used to solve many polynomial equations and derive their factors. Since the polynomial equations are tough to solve, Biquadratic Equations were formed to make the process easy. All Biquadratic equations are polynomial equations, but all polynomial equations are not biquadratic equations. A basic understanding of this equation is that if an equation has an odd-degree term, this substitution is not applicable. Biquadratic Substitutions form an integral part of the theories of indefinite integration that includes quadratic and biquadratic equations. 

What are biquadratic substitutions and a Biquadratic Equation?  

In standard terms, the algebraic equation is known as a quadratic equation. The primary application to biquadratic equations is such that it cannot be applied to the equations having odd-degree. A polynomial equation can be converted into a biquadratic equation or forming part of Biquadratic Substitutions. 

The technique used in the process of this conversion is the supposition technique. In this technique, a variable with a certain degree can be converted to another variable. The supposition takes place so that the equation is transformed into a quadratic equation. Biquadratic equations have been studied for a long time. There are two kinds of algebraic methods used to solve the equations.

They are Ferrari’s Solution and Descarte’s Solution. Under Ferrari’s Solution, each polynomial equation with a leading coefficient with value 1 and its degree is an odd number and has at least one real root, and has a real root that has an opposite sign to its last term. 

Under Descarte’s solution, there is no possibility of the existence of a general formula that is expressed in algebraic operations. Such operations form part of the polynomial equations for the roots of the equation if the degree of the equation is higher than 4. This might help derive the solution for the question “What is Biquadratic Substitution?”

How to Solve any Biquadratic Equation?

There are two methods used to solve the Biquadratic Equation. The Biquadratic Substitutions also form part of these theories. They are Ferrari’s solution and Descarte’s solution. 

Ferrari’s solution can be understood with the help of the below-mentioned examples. 

a4 – 2a3 – 5a2 + 10a – 3 = 0

Here, the quadratic polynomial can be added to both sides. The next step has to be done so that a and b are chosen so that the left-hand side of the number becomes a perfect square. When the value of k is unknown, a and b have to be chosen. After this, we arrive at a cubic equation called the resolvent cubic of the biquadratic solution given. This was obtained by eliminating both a and b from the equation. Here either one of the cubic roots is chosen. 

Descarte’s solution can be understood with the help of the following equation:

Here, we are supposed to solve the quadratic equations to derive the 4 roots of the original quartic. In this expression: 

a4 – 2a3 – 5a2 + 10a – 3 = 0

The cube term has to be removed, and the equation is further derived. The left-hand side becomes the product of quadratic polynomials. While solving these equations, we must keep in mind that the coefficients of y in all of these factors are k and -k, as the product does not contain any of the terms with y3. 

Now this equation has one real root and a positive root. 

This gives a clear understanding to the question “What is Biquadratic Substitution?”

What are Quadratic Equations?  

Quadratic equations are an equation that can be rearranged in their standard form. It is a second-order polynomial equation that comprises a single variable x a+2+bx+c=0. Here, a is not equal to zero. Since a quadratic equation forms part of second-order polynomial equations, it is guaranteed to have at least one solution. And this solution can either be real or complex. 

Distinctions Between Biquadratic and Quadratic Equations

The basic distinctions between biquadratic and quadratic equations are such that a quartic relates to the fourth degree. In contrast, biquadratic are polynomial expressions that involve the 2nd and 4th powers of the variables. The other difference is that the quadratic is an algebraic equation or the fourth terms’ function while the other is biquadratic. Also, the quadratic are square-shaped, while the biquadratic are polynomial expressions with the 2nd and 4thpowers of a variable. The equations of quartic have the general form as ax2 + bx + c = 0 whereas the biquadratic equation has its general form as ax4 + hx3 + cx2 + dx + e = 0. A standard quadratic equation can also be written as ax4 + hx3 + cx2 + dx + e = 0.  

A lot of Biquadratic Substitution formulas form part of this theory. 

Conclusion 

There are a lot of common mistakes that are made in Biquadratic Substitutions that have to stand corrected while dealing with the polynomial equations. Some of these mistakes include the dropping of ± while taking roots of the factors, etc. Generally, in the process of studying the concepts of Biquadratic Substitutions under the theories of polynomial equations, we might come across a lot of product and quotient rules.