Algebraic Identity (a - b)²

Objective

To verify the algebraic identity (a-b)² = a² - 2ab + b².

Algebraic identity (a-b)²

  • An algebraic identity is an equality, which is true for all values of the variables in the equality.The algebraic equations which are valid for all values of variables in them are called algebraic identities.For example x²-4=(x+2)(x-2) is an identity.
     
  • On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.For example, x+3=7, is an equation.

1. Linear Operation Comparison:

  • Linear operations follow the properties of linearity, such as f(x + y) = f(x) + f(y).
  • Examples include addition, differentiation, and complex conjugation.

2. Squaring of (a - b):

  • Start with the expression (a - b)2.
  • Expand it as (a - b)(a - b).
  • Apply the distributive property to get a2 - 2ab + b2.

3. Mixed Product Term:

  • Notice the term 2ab in the result.
  • This term involves the product of both a and b, introducing non-linearity.

4. Non-Linearity Consequence:

  • Because of the mixed product term, squaring does not satisfy the linearity property.
  • Unlike linear operations, the squaring operation has this additional term, making it nonlinear.

5. Example:

  • Let's take a = 3 and b = 2 for illustration.
  • (3 - 2)2 = 12 = 1.
  • However, if squaring were a linear operation, we would expect (3 - 2)2 to be equal to 32 - 22, which is not the case.

6. Comparison with Linear Operations:

  • Linear operations like addition or differentiation would have yielded (3 - 2)2 = 32 - 2(3)(2) + 22, which simplifies to 9 - 12 + 4 = 1.

7. Conclusion:

  • Understanding the non-linearity of squaring is important in various mathematical contexts, as it introduces unique characteristics and consequences not found in linear operations.
     

Pre-requisite Knowledge

Square

A square is a regular quadrilateral, which means that it has four equal sides and four equal angles of 90 degrees each.

Area of a Square

  • The product of the length of each side itself.
  • Formula: Area = side²

Rectangle

Rectangle is a parallelogram with four right angles and opposite sides are equal. The length of a rectangle is the size of the longer side, whereas the breadth is the size of the shorter side.

Area of a Rectangle

  • The product of its length and breadth
  • Formula: Area = length x breadth