Know about Euler’s dynamical equations

Algebraic Equations: An Introduction 

In the field of Mathematics, an algebraic equation can be defined as a statement where an algebraic expression is equal to another. These expressions are formed by using algebraic operations like addition, subtraction, multiplication, division, raising to a power and extraction of a root to variables and constants. In modern algebra, algebraic equations may also be formed by applying non-algebraic operations, such as logarithms or trigonometry to variables and constants and equalizing two expressions. 

We can solve an algebraic equation to find a particular number or a set of number, which when substituted in place of the variables, reduces the equation to a true identity. The numbers or values that satisfy a given equation are often called roots of the equation. This root may be real or imaginary, depending upon the equation.

Here are a few examples of algebraic equations. 

3×3+4y3+5×2=32

tan(x)+sec (y) = 13

x2-x-1=0

Types of Algebraic Equations

Depending on the complexity and the data to be calculated in a problem, several kinds of algebraic equations can be formulated. There are five main classifications of algebraic equations that are used in the field of Mathematics.

Polynomial Equations: Equations with whole number exponents for every variable are known as polynomial equations. When polynomial equations have a degree of 2, it is known as a quadratic algebraic equation or simply a quadratic equation and when the degree is 3, it is a cubic algebraic equation or cubic equation. Given below are a few examples of polynomial equations. 

2x=9

4×4+7×2+9x-4=0

Exponential Equations: These equations have a variable term in their exponents. An example would be

3(x-4)=9.

Logarithmic Equations: Logarithmic equations are the inverse of exponential equations. For example, if an equation in form of exponents is given by 7(2x-3)=49, then the same equation in logarithmic form is given by log749=2x-3.

Rational Equations:

Rational equations are algebraic equations which are in the form of p(x)q(x), where p(x) and q(x) are both polynomials and q(x)0. Here is an example of rational equation, 

7x-2×2+9=92

Trigonometric equations:

These are equations that contain the trigonometric function sin, tan, cos, sec, cosec and cot. Given below is an example of a trigonometric equation.

tan(x)=2

Solving Quadratic Equations

A quadratic equation is a type of polynomial equation, where the maximum power a variable can have is 2. Here is a quadratic equation in its general form. 

ax2+bx+c=0, where

In this equation, a is the coefficient of x2, b is the coefficient of x and c is a constant. 

Usually, quadratic equations are solved using identities, factorization, long division, splitting the middle term and using graphs, among several other methods. In all cases, quadratic equations will have two roots or solutions. However, they may both be equal, giving that feel that there is only one root. 

The solution to the general form of a quadratic equation can be given by the following equation.