Question

What is an algebraic equation?

Answer 1

Hint: First we will understand the definition of an algebraic expression using examples. Now, we will understand how an algebraic equation is formed from an algebraic expression. Further will see different types of equations like: - linear equation, quadratic equation, cubic equation, bi-quadratic equation etc. with some examples.

Complete step-by-step solution:
Here we have been asked to describe the term algebraic equation. But first we need to understand the term ‘algebraic expression’. Let us understand them using some examples.
In mathematics, an algebraic expression is an expression that contains constants, variables and algebraic operations like: - addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number. For example: - $2x+1$, ${{x}^{2}}+x+1$, ${{x}^{3}}+2{{x}^{2}}+5x+3$ etc. are algebraic expressions.
Now, If we will substitute an algebraic expression equal to 0 or some constant and then by simplifying it we can write it in the form $f\left( x \right)=0$ for the variable x then we will get an algebraic equation. For example: - $2x+1=0$, ${{x}^{2}}+x+1=0$, ${{x}^{3}}+2{{x}^{2}}+5x+3=0$ etc. The nature of the algebraic equation depends on the highest exponent of the variable present in the equation. If the highest exponent of the variable is 1 then it is called linear equation, for 2 it is called a quadratic equation, for 3 it is called a cubic equation and for 4 it is called a bi – quadratic equation and so on.

Note: Note that it may be possible that we have a multivariate algebraic expression. Multivariate means there may be more than one variable in the expression. For example: - ${{x}^{3}}{{y}^{2}}+2{{x}^{2}}z+5{{y}^{3}}z+3$, here we have three variables x, y and z. In this case we generally prefer to use the term ‘polynomial expression’ instead of algebraic expression and similarly if we will write ${{x}^{3}}{{y}^{2}}+2{{x}^{2}}z+5{{y}^{3}}z+3=0$ then it will be termed as polynomial equation and not an algebraic equation.