a² + b² and a² - b² Formulas: Complete Guide with Examples

a² + b² and a² - b² Formulas: In this guide, we explore the essential algebraic identities for a² + b² and a² - b² with practical examples. These algebraic expressions are fundamental Algebra formulas required for high school mathematics and competitive examinations, including the NTSE, NDA, AFCAT, SSC, and Railway exams. Mastering these basic identities allows you to solve complex mathematical problems with speed and precision.

a2 b2 Formula a²+b²

Let variables 'a' and 'b' represent any two algebraic terms. When we sum the squares of these two terms, it is expressed as a²+b². This binomial expression is a cornerstone of algebra. Below, we detail the derivations and applications of the a²+b² formula.

Deriving the a²+b² Formula: 

We know from the expansion of binomial squares that (a + b)² = a² + b² + 2ab

By rearranging the terms, we get: a² + b² = (a + b)² - 2ab

Similarly, considering the square of a difference: (a - b)² = a² + b² - 2ab

Rearranging this yields: a² + b² = (a - b)² + 2ab 

Consequently, there are two common variations of the sum of squares formula, as summarized below:

1. a² + b² = (a + b)² - 2ab

2. a² + b² = (a - b)² + 2ab 

a2 b2 Formula a²-b²

Next, consider the difference of squares, denoted as a²-b². This represents a fundamental binomial factorization. The formula for a²-b² is essential for simplifying algebraic expressions and solving equations, as explained below.

The a²-b² Formula: 

The factors of a²-b² are (a + b) and (a - b). This identity holds great significance and can be easily visualized through geometric derivation. 

Imagine a large square with side length 'a'. If you subtract a smaller square with side length 'b' from it, the remaining area is defined as a²-b². This geometric shape can be rearranged into a rectangle with length (a + b) and width (a - b). Since the area of the rectangle is equivalent to the area of the remaining square, we can conclude: 

a² - b² = (a + b) (a − b) 

a2 b2 Formula with Examples

Example 1: Using the sum of squares formula, calculate the value of 9² + 12².

Solution: Given a = 9 and b = 12.

Applying the a² + b² formula:

a² + b² = (a + b)² - 2ab

9² + 12² = (9 + 12)² - 2(9)(12) 

9² + 12² = (21)² - 216 

9² + 12² = 441 - 216 = 225

Example 2: Find the value of the expression 3² + 5² using the sum of squares formula.

Solution: Given a = 3 and b = 5.

Using the sum of squares identity:

a² + b² = (a + b)² − 2ab

3² + 5² = (3 + 5)² - 2(3)(5) = 64 - 30

3² + 5² = 34

Example 3: Prove that x² + y² = (x + y)² - 2xy using the a² + b² formula.

Solution: To prove x² + y² = (x + y)² - 2xy, we start with the standard expansion.

By the identity:

(a + b)² = a² + b² + 2ab

Substitute a = x and b = y:

After expanding and re-arranging terms:

(x + y)² = x² + y² + 2xy

Subtracting 2xy from both sides, we reach the required expression:

x² + y² = (x + y)² - 2xy (Hence Proved)

Example 4: Calculate the difference of squares 12² - 4² using the a² - b² formula. 

Solution: Given a = 12 and b = 4.

Applying the identity:

a² - b² = (a + b) (a - b)

12² - 4² = (12 + 4) (12 - 4) = 16 × 8 = 128

Example 5: Solve (13 + 6)(13 - 6) using the subtraction of squares formula.

Solution: Here, a = 13 and b = 6.

Applying the formula:

a² - b² = (a + b) (a - b)

13² - 6² = (13 + 6) (13 - 6)

169 - 36 = 133