Algebraic Equations
Algebraic equations are two algebraic expressions that are joined together using an equal to ( = ) sign. An algebraic equation is also known as a polynomial equation because both sides of the equal sign contain polynomials. An algebraic equation is built up of variables, coefficients, constants as well as algebraic operations such as addition, subtraction, multiplication, division, exponentiation, etc.
If there is a number or a set of numbers that satisfy the algebraic equation then they are known as the roots or the solutions of that equation. In this article, we will learn more about algebraic equations, their types, examples, and how to solve algebraic equations.
| 1. | What is Algebraic Equations? |
| 2. | Types of Algebraic Equations |
| 3. | Algebraic Equations Formulas |
| 4. | How to Solve Algebraic Equations |
| 5. | FAQs on Algebraic Equations |
What is Algebraic Equations?
An algebraic equation is a mathematical statement that contains two equated algebraic expressions. The general form of an algebraic equation is P = 0 or P = Q, where P and Q are polynomials. Algebraic equations that contain only one variable are known as univariate equations and those which contain more than one variable are known as multivariate equations. An algebraic equation will always be balanced. This means that the right-hand side of the equation will be equal to the left-hand side.
Algebraic Expressions
A polynomial expression that contains variables, coefficients, and constants joined together using operations such as addition, subtraction, multiplication, division, and non-negative exponentiation is known as an algebraic expression. An algebraic expression should not be confused with an algebraic equation. When two algebraic expressions are merged together using an "equal to" sign then they form an algebraic equation. Thus, 5x + 1 is an expression while 5x + 1 = 0 will be an equation.
Algebraic Equations Examples
x2 - 5x = 3 is a univariate algebraic equation while y2x - 5z = 3x is an example of a multivariate algebraic equation.
Types of Algebraic Equations
Algebraic equations can be classified into different types based on the degree of the equation. The degree can be defined as the highest exponent of a variable in an algebraic equation. Suppose there is an equation given by x4 + y3 = 35 then the degree will be 4. In determining the degree, the exponent of the constant or coefficient is not considered. The number of roots of an algebraic equation depends on its degree. An algebraic equation where the degree equals 5 will have a maximum of 5 roots. The various types of algebraic equations are as follows:
Linear Algebraic Equations
A linear algebraic equation is one in which the degree of the polynomial is 1. The general form of a linear equation is given as a1x1+a2x2+...+anxn = 0 where at least one coefficient is a non-zero number. These linear equations are used to represent and solve linear programming problems.
Example: 3x + 5 = 5 is a linear equation in one variable. y = 2x - 6 is a linear equation in two variables.
Quadratic Algebraic Equations
An equation where the degree of the polynomial is 2 is known as a quadratic algebraic equation. The general form of such an equation is ax2 + bx + c = 0, where a is not equal to 0.
Example: 3x2 + 2x - 6 = 0 is a quadratic algebraic equation. This type of equation will have a maximum of two solutions.
Cubic Algebraic Equations
An algebraic equation where the degree equals 3 will be classified as a cubic algebraic equation. ax3 + bx2 + cx + d = 0 is the general form of a cubic algebraic equation (a ≠ 0).
Example: x3 + x2 - x - 1 = 0. A cubic algebraic equation will have a maximum of three roots as the degree is 3.
Higher-Order Polynomial Algebraic Equations
Algebraic equations that have a degree greater than 3 are known as higher-order polynomial algebraic equations. Quartic (degree = 4), quintic (5), sextic (6), septic (7) equations all fall under the category of higher algebraic equations. Such equations might not be solvable using a finite number of operations.
Algebraic Equations Formulas
Algebraic equations can be simplified using several formulas and identities. These help to expedite the process of solving a given equation. Given below are some important algebraic formulas:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)(a - b) = a2 - b2
- (x + a)(x + b) = x2 + x(a + b) + ab
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a - b)3 = a3 - 3a2b + 3ab2 - b3
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- Quadratic Formula: [-b ± √(b² - 4ac)]/2a
- Discriminant: b2 - 4ac
How to Solve Algebraic Equations
There are many different methods that are available for solving algebraic equations depending upon the degree. If an algebraic equation has two variables then two equations will be required to find the solution. Thus, it can be said that the number of equations required to solve an algebraic equation will be equal to the number of variables present in the equation. Given below are the ways to solve algebraic equations.
Linear Algebraic Equations
A linear algebraic equation in one variable can be solved by simply applying basic arithmetic operations on both sides of the equation.
E.g: 4x + 1 = 5.
4x = 5 - 1 (Subtracted 1 from both sides).
4x = 4 (Solve the R.H.S using algebraic operations)
x = 1 (Divided both sides by 4)
Linear algebraic equations in more than one variable will be solved using the concept of simultaneous equations.
Quadratic Algebraic Equations
A quadratic algebraic equation can be solved by using identities, factorizing, long division, splitting the middle term, completing the square, applying the quadratic formula, and using graphs. A quadratic equation will always have a maximum of two roots.
E.g: x2 + 2x + 1 = 0
Using the identity (a + b)2 = a2 + 2ab + b2, we get
a = x and b = 1
(x + 1)2 = 0
(x + 1)(x + 1) = 0
x = -1, -1.
The most effective way of solving higher-order algebraic polynomials in one variable is by using the long division method. This decomposes the higher-order polynomial into polynomials of a lower degree thus, making it easier to find the solutions.
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Important Notes on Algebraic Equations:
- An algebraic equation is an equation where two algebraic expressions are joined together using an equal sign.
- Polynomial equations are algebra equations.
- Algebraic equations can be one-step, two-step, or multi-step equations.
- Algebra equations are classified as linear, quadratic, cubic, and higher-order equations based on the degree.
FAQs on Algebraic Equations
What are Algebraic Equations?
Algebraic equations are polynomial equations where two algebraic expressions are equated. Both sides of the equation must be balanced. The general form of an algebraic equation is P = 0.
What is an Example of Algebraic Equation?
An algebraic equation can be linear, quadratic, etc. Hence, an example of an algebraic equation can be 3x2 - 6 = 0.
How Do You Solve Algebraic Equations?
There are many methods available to solve algebraic equations depending on the degree. Some techniques include applying simple algebraic operations, solving simultaneous equations, splitting the middle term, quadratic formula, long division, and so on.
What are Algebraic Expressions and Algebraic Equations?
Mathematical statements that consist of variables, coefficients, constants, and algebraic operations are known as algebraic expressions. When two algebraic expressions are equated together, they are known as algebraic equations.
How Do You Write Algebraic Equation?
We can convert real-life statement involving numbers and conditions into algebraic equation. For example, if the problem says, "the length of a rectangular field is 5 more than twice the width", then it can be written as the algebraic equations l = 2w + 5, where 'l' and 'w' are the length and width of the rectangular field.
What are Linear Algebraic Equations?
An algebraic equation where the highest exponent of the variable term is 1 is a linear algebraic equation. In other words, algebraic equations with degree 1 will be linear. For example, 3y - 9 = 1
Are Quadratic Equations Algebraic Equations?
Yes, quadratic equations are algebraic equations. It consists of an algebraic expression of the second degree.
What are the Basic Formulas of Algebraic Equations?
Some of the basic formulas of algebraic equations are listed below:
- (a + b)2 = a2 + 2ab + b2
- (a - b)2 = a2 - 2ab + b2
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a - b)3 = a3 - 3a2b + 3ab2 - b3
- Quadratic Formula: [-b ± √(b² - 4ac)]/2a
- Discriminant: b2 - 4ac
What are the Rules for Algebraic Equations?
There are 5 basic rules for algebraic equations. These are as follows: