ALGEBRAIC FORMULAE 

 $$ \begin{aligned} & (a+b)^2=a^2+2 a b+b^2 \\ & (a-b)^2=a^2-2 a b+b^2 \\ & a^2+b^2=(a+b)^2-2 a b \\ & a^2-b^2=(a-b)(a+b) \\ & (a+b)^3=a^3+3 a^2 b+3 a b^2+b^3 \\ & (a-b)^3=a^3-3 a^2 b+3 a b^2-b^3 \\ & a^3-b^3=(a-b)\left(a^2+a b+b^2\right) \\ & a^3+b^3=(a+b)\left(a^2-a b+b^2\right) \\ & (a+b+c)^2=a^2+b^2+c^2+2 a b+2 b c+2 c a \\ & (a-b-c)^2=a^2+b^2+c^2-2 a b+2 b c-2 c a \end{aligned} $$

An algebraic formula is a mathematical expression involving variables, numbers, and operations like addition, subtraction, multiplication, and division. It represents a general relationship or rule, allowing for the substitution of specific values. Algebraic formulas are crucial in various fields for modeling, analyzing, and solving problems.

Importance:
1. **Problem Solving:** Formulas provide systematic approaches to solving problems in mathematics and real-world applications, aiding in finding solutions efficiently.

2. **Modeling Relationships:** Formulas express relationships between variables, enabling the modeling of complex situations, such as economic trends, scientific laws, or engineering principles.

3. **Consistency and Precision:** Algebraic formulas provide a precise and consistent way to communicate mathematical relationships, reducing ambiguity and facilitating clear communication.

4. **Generalization:** Formulas allow the generalization of patterns and relationships, making it possible to derive solutions for a wide range of specific cases.

5. **Scientific Research:** In scientific research, formulas are essential for expressing laws, theories, and mathematical relationships that help understand and predict phenomena.

6. **Engineering and Technology:** Formulas play a crucial role in engineering design, optimization, and analysis, providing the mathematical foundation for constructing solutions in various technological fields.

7. **Financial Planning:** Formulas are used in finance for calculations related to interest rates, investments, and financial planning, aiding in decision-making processes.

8. **Computer Science:** Algorithms and computations heavily rely on algebraic formulas, forming the basis for programming and data manipulation.

In summary, algebraic formulas are fundamental tools that offer a systematic way to represent and analyze relationships, making them indispensable across diverse fields of study and application.