An algebraic equation is a mathematical statement that expresses the equality of two expressions by connecting them with an equals sign (=). It contains variables, constants, and algebraic operations (addition, subtraction, multiplication, division, etc.) which are used to show relationships between different things.
From simple tasks like splitting a bill to bigger challenges like building bridges or exploring science, algebraic equations play a key role.
Some Examples of Equations
- 2x + 3y = 7 ( Linear Equation )
- x2 + 4x + 5 = 0 ( Quadratic Equation )
- 3x3 + 4x + 7 = 0 ( Cubic Equation )
Degree of Algebraic Equations
The degree of an algebraic equation is the highest power (exponent) of the variable in the equation when it is expressed in its standard form. The degree determines the number of roots or solutions the equation can have. An algebraic equation can have n number of solutions where n is the degree of that algebraic equation.
- A linear equation of degree 1 only has a single solution.
- A quadratic equation of degree 2 can have two solutions.
Types of Algebraic Equations
There are many different forms of algebraic equations based on their structure and the number of variables involved. Based on their degrees the algebraic equations can be divided into mainly four categories:
- Linear Equations
- Quadratic Equations
- Cubic Equations
- Higher Order Algebraic equations
Linear Equation
A linear equation is an equation in which the highest power of a variable is 1. They are also known as first-order equations. These equations are the simplest type and represent a straight line when graphed.
The general form of linear equation is represented as:
a1x1 + a2x2 + ...+ anxn + c = 0
where x1 , x2 , x3 , x4 , x5 , ..... xn are the variables and a1 , a2 , a3....an are the coefficients of the variables at least one coefficient is a non-zero number.
Some Examples of linear Equation:
- ax + b = 0 ( Linear equation in one variable( x) )
- ax + by + c = 0 ( Linear Equation in two variable ( x, y ) )
Representation of Linear equationQuadratic Equation
A quadratic Equation is a type of algebraic equation in which the highest power of a variable is 2. They are also known as second-degree equation and it forms a U-shaped curve called a parabola when plotted on a graph.
The general form of quadratic equation is:
ax2 + bx + c = 0,
where a, b, and c are constants and x is the variable.
Some examples of quadratic equations are :
- 3x2 + 5x + 7 = 0
- y = 4x2 + 2x + 6
A quadratic equation can have two solutions which can be either imaginary or real depending on the equation.
Quadratic EquationCubic Equation
A cubic Equations is a type of equation where the highest power of the variable is 3. They are also known as third- degree equations. and they form an "S" or "N" shape on the graph, with up to three points where it crosses the x-axis .
A cubic equation has the general form :
ax3 + bx2 + cx + d = 0.
where x is a variable and a, b, c, and d are constants.
A cubic equation can have one, two, or three solutions that are real or complex numbers, depending on the coefficients in the equation.
Some examples of Cubic Equations are :
- 3x3 + x2 + 4x + 9 = 0
- 7x3 + 5x + 3 = 0
Representation of a Cubic EquationHigher-Order Algebraic Equations
Higher-order algebraic equations are equations where the highest power of the variable (called the degree) is greater than three. These equations go beyond linear (x1), quadratic (x2), and cubic (x3) equations, involving degrees such as 4, 5, or even higher.
Some Examples of Higher-Order Equations:
- Quartic Equation (Degree 4): x4 + 2x3 − x2 + 5 = 0
- Quintic Equation (Degree 5): 2x5 − 3x3 + x2 − 4 = 0
How To Solve Algebraic Equations
Solving linear Equations
- Linear equation in one variable :
To solve a linear equation in one variable, isolate the variable by performing inverse operations on both sides of the equation while keeping it balanced. Simplify step by step until the variable is alone.
- Linear equation in two variable :
- Substitution Method:
Solve one equation for one variable in terms of the other.
Substitute this expression into the second equation and solve for the remaining variable. - Elimination Method (or Addition/Subtraction Method):
Add or subtract the equations to eliminate one variable.
Solve for the remaining variable, then substitute back to find the other.
Solving Quadratic Equations
- Quadratic Equations: There are four main methods to solve quadratic equations:
- Factoring: Split the equation into factors and solve for the variable.
- Quadratic Formula: Use the formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
- Completing the Square: Rewrite the equation into a perfect square form and solve.
- Graphing: Plot the equation and find the points where it crosses the x-axis.
Read More:
- (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