a2 b2 Formula and Examples

a2 b2 Formula: Here we shall discuss the a2 b2 algebraic formula like a²+b² and a²-b² with some examples. The a2 b2 formula is one of the most basic Algebra formulas that is primarily used in mathematical calculations in class from 10th onwards and in competitive exams like NTSE, NDA, AFCAT, SSC, Railways, etc. It is always advisable to memorize these basic algebra formulas so that you can find the solution to mathematical problems quickly and easily.

a2 b2 Formula a²+b²

Let us consider that a and b are two mathematical variables which denote 2 terms of algebra. When you add the square of both algebraic terms, it will be written as a²+b². It expresses a binomial algebraic equation. The a2 b2 formula of a²+b² is explained below with the mathematical expressions.

For the a²+b² Formula, 

As we know that (a +b)² = a² + b² + 2ab

a² + b² = (a +b)² - 2ab

Similarly, As we know that (a -b)² = a² + b² - 2ab

a² + b² = (a -b)² + 2ab 

So, there are two formulas related to a²+b² as mentioned below.

1. a² + b² = (a +b)² - 2ab

2. a² + b² = (a -b)² + 2ab 

a2 b2 Formula a²-b²

Let us consider that a and b are two mathematical variables which denote 2 terms of algebra. When you subtract the square of both algebraic terms, it will be written as a²-b². It expresses a binomial algebraic equation. The a2 b2 formula of a²-b² is explained below with the mathematical expressions.

For the a²-b² Formula, 

The factors of a²-b² are the (a+b) and (a-b). The a²-b² Formula can be derived geometrically. 

After subtracting the part of a small square-shaped box of side b from a big square of side a then a new geometric shape is produced with its area a²-b².  It is found that the area of the subtracted shape is a²-b² and the same shape can be transformed as a rectangle with the length of (a+b) and width of (a-b). As the area of that rectangle is equivalent to the area of the square. So, we can get that 

a²-b² = (a+b) (a−b) 

a2 b2 Formula with Examples

Question 1: With the help of the sum of squares formula, calculate the value of (9)² + (12)².

Solution: Given that the value of a = 9, b = 12

By using the a² + b² Formula,

a² + b² = (a +b)² - 2ab

9² + 12² = (9 + 12)² - 2 (9)(12) 

9² + 12² = 21² - 2 (9) (12) 

9² + 12² = 441- 216 = 225

Question 2: With the help of the sum of squares formula, find the value of the given expression 3² + 5².

Solution: Given that the value of a = 3, b = 5

By using the sum of squares formula,

a² + b² = (a + b)² − 2ab

3² + 5² = (3 + 5)² - 2 (3) (5)

3² + 5² = 64 - 30 = 34

Question 3: Verify with the help of the a² + b² formula that the value of the given expression x² + y² is (x + y)² - 2xy.

Solution: As per the question, to verify x² + y² = (x + y)² - 2xy by using the a² + b² formula.

By using the a² + b² formula,

(a + b)² = a² + b² + 2ab

Here, a = x, b = y

After expanding and substituting the algebraic terms,

(x + y)² = x² + y² + 2xy

We get the required expression,

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

Question 4: Calculate the subtraction of the squares of 12 and 4 directly by using the a² - b² formula. 

Solution: Given that the value of a = 12, b = 4

By using the subtraction of squares formula,

a² - b² = (a + b) (a - b)

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

Question 5: With the help of the subtraction of squares formula, find the value of the given expression (13 + 6) (13 - 6).

Solution: Given that the value of a = 13, b = 6

By using the subtraction of squares formula,

a² - b² = (a + b) (a - b)

13² - 6² = (13 + 6) (13 - 6)

(13 + 6) (13 - 6) = 169 - 36 = 133