What is a2 b2 formulas? Proof and Questions and Answers

a2 b2 Maths Formula

Let's consider $a$ and $b$ as two mathematical variables representing algebraic terms. The a2+b2 formula is used calculate the sum of two or more squares within an expression. By solving  the (a+b)²or (a-b)², we can efficiently derive the a2 + b2 formula. In essence, the sum of squares formula, commonly known as the a2 +b2 formula, can be expressed as follows: We are aware of this. (a +b)² = a² + b² + 2ab a² + b² = (a +b)² – 2ab Also we can say that (a -b)² = a² + b² – 2ab a² + b² = (a -b)² + 2ab The two formulas for a2+b2 are

a² + b²

a² + b² = (a +b)² – 2ab a² + b² = (a -b)² + 2ab

Steps for Applying the A2 + B2 Formula in math

The sum of squares formula, or a2 + b2 formula, is applied in the following steps:

Step 1 Identify Variables: Determine the values of $a$ and $b$ in the expression a² + b².

Step 2 Square each term: Square each value of $a$ and $b$ individually. This involves multiplying each value by itself.

Step 3 Add Squares: Add the squared values of $a$ and $b$ together.

Step 4 Apply Formula: The sum of squares formula a² + b² is now applied.

Step 5 Simplify: Simplify the expression further if possible by combining like terms or using algebraic techniques.

Step 6 Finalize: Express the result in its final form, which may include numerical values or variables depending on the context of the problem.

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a2-b2 Formula

The formula a2-b2 is commonly known as "the difference of squares formula." It is used to find the difference between two squares without having to compute the squares individually. The a2-b2 formula is particularly useful for factoring square binomials. The a2-b2 formula is represented as: a2 – b2 = (a – b) (a + b $)$ .

a2-b2 Formula- Proof

To demonstrate that a2 – b2 = (a – b) (a + b), we must prove LHS = RHS. Let us try to solve the following equation: a2 – b2 = (a – b) (a + b) Multiplying  (a – b) and (a + b) we get, =a(a+b) -b(a + b) =a2 + ab – ba – b2 =a2 + 0 + b2 =a2 – b2 . Hence we can say that a2 – b2 = (a – b) (a + b).

Examples on a2 – b2 Formula

Example 1: Simplify x ² – 16

Solution:

= x ² – 16

= x ² – 4 ²

We know that, a ² – b ² = (a+b) (a–b)

Given,

• a = x
• b = 4

= (x + 4)(x – 4)

Example 2: Using the a² +b² formula, calculate the sum of 14² + 20²

Solution:

Here.the value of a = 14 and b= 20 .

The formula of the sum of a²b² formula is

(a²+b²)= (a+b)²-2ab

=  (14²+2×14×20+20²)- 2×14×20 [using (a+b)² formula]

= 196 + 560+400 -560

=196 +400 = 596.

Example 3: Simplify (3x + 2) ² – (3x – 2) ²

Solution:

We know that,

a ² – b ² = (a+b)(a–b)

Given,

• a = 3x + 2
• b = 3x – 2

(3x + 2) ² – (3x – 2) ²

= (3x + 2 + 3x – 2)(3x + 2 – (3x – 2))

= 6x(3x + 2 – 3x + 2)

= 6x(4)

= 24x

Example 4: Find the value of 100²-8², using the a²-b² formula.

Solution:

The formula of the Subtraction of squares or a²- b² formula is

(a²- b²) =(a+b)(a-b)

In the given expression, a=100 ,b= 8

100²-8²= (100+8)(100-8)

= 108× 92 =9936