# Algebraic Equations

**Algebraic Equations**

An algebraic equation is a mathematical statement that uses letters or symbols to represent numbers. These letters are typically variables, and the equation relates these variables using mathematical operations like addition, subtraction, multiplication, division, and exponentiation. The goal is to find the values of the variables that satisfy the equation, making it true.

**1. Types of Algebraic Equations**

a. **Linear Equations:** - These equations have the highest power of the variable as 1. - General form: ax + b = 0 - Example: 2x + 5 = 0

b. **Quadratic Equations:** - These equations have the highest power of the variable as 2. - General form: a${x}^{2}$ + bx + c = 0 - Example: 3${x}^{2}$ - 4x + 1 = 0

c. **Cubic Equations:** - These equations have the highest power of the variable as 3. - General form: a${x}^{3}$ + b${x}^{2}$ + cx + d = 0 - Example: ${x}^{3}$ + 2${x}^{2}$ - 4x - 8 = 0

d. **Quartic Equations:** - These equations have the highest power of the variable as 4. - General form: a${x}^{4}$ + b${x}^{3}$ + c${x}^{2}$ + dx + e = 0 - Example: 2${x}^{4}$ - 5${x}^{3}$ + 3${x}^{2}$ + 6x - 1 = 0

**2. Solving Algebraic Equations**

a. **Isolating the Variable:** - The first step is to isolate the variable on one side of the equation. Use inverse operations (addition, subtraction, multiplication, division) to do this.

b. **Balancing Equations:** - Perform the same operations on both sides of the equation to keep it balanced.

c. **Substitution:** - In some cases, you may need to substitute one expression for another to simplify the equation.

**3. Common Techniques for Solving Equations**

a. **Factoring:** - Involves breaking down the equation into its factors to find the values of the variables.

b. **Quadratic Formula:** - Used to solve quadratic equations: x = (-b ± √(${b}^{2}$ - 4ac)) / 2a

c. **Completing the Square:** - Used to convert a quadratic equation into a perfect square trinomial.

d. **Graphical Method:** - Plot the equation on a graph and find the intersection with the x-axis to locate solutions.

e. **Matrix Methods:** - Used for solving systems of linear equations.

**4. Special Equations**

a. **Absolute Value Equations:** - Equations involving the absolute value of a variable.

b. **Rational Equations:** - Equations where the variable is in the denominator.

c. **Exponential Equations:** - Equations with variable exponents.

**5. Word Problems:**

- Many real-life problems can be modeled and solved using algebraic equations. Translate the problem into an equation, solve it, and interpret the results in the context of the problem.

**6. Inequalities:**

- Algebraic equations can also be used to represent inequalities. The solution to an inequality is a range of values that satisfy the inequality.