A² + B² and A² – B² Formulas: Definition, Proof, Derivation & Examples

A² + B² and A² – B² Formulas: In algebra, the expressions a² + b² and a² – b² are two important identities that help students to simplify complex problems (these identities are helpful at the school level and also after).

The formula a² + b² shows the sum of the squares of two numbers, while a² – b² represents the difference of their squares. These A² + B² and A² – B² formulas form the foundation for solving algebraic identities, factoring, and simplifying expressions. 

These A² + B² and A² – B² formulas are widely used in school-level mathematics, basically in 6th to 12th standard, board exams (CBSE, ICSE, State Boards), as well as competitive exams such as SSC, Banking, Railways, JEE, and Olympiads. Understanding these formulas also helps in geometry (e.g., Pythagoras theorem) and physics calculations. Let us now understand both formulas in an easy and detailed way.

What Is the A² + B² Formula?

The expression a² + b² shows the sum of the squares of two numbers, a and b. This identity is commonly derived using the expansion of (a + b)² and (a – b)².

We know that:

• (a + b)² = a² + b² + 2ab

• (a – b)² = a² + b² – 2ab

Therefore, by rearranging the terms of the equation:

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

Standard Identity used everywhere:

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

Steps to Apply the A² + B² Formula

Here is how to apply the formula logically:

1. Identify the values of 'a' and 'b' in any question that you are solving.

2. Square each of the given terms (i.e., find a² and b²).

3. Add the results of the equation to get a² + b².

4. Alternatively, use the identity: a² + b² = (a + b)² – 2ab.

5. Simplify to get the final answer. For example, if any question is given, just step by step solve the whole equation, you will get the answer in the end.

 Example of this type of Equation:

Find the value of 6² + 8². Solve in Detail.

Solution:
a = 6, b = 8 i.e., See in this question the value of a and b is given, and like this in every question the values are given, students just need to put the values and solve the question.
a² + b² = 6² + 8² = 36 + 64 = 100 

What Is the A² – B² Formula?

The expression a² – b² is known as the difference of squares. Minus represents the difference in Maths. For example, (of) represents multiplication in maths.

 Standard Identity used in Mathematics:

a² – b² = (a + b)(a – b)

Why is this identity useful in Maths?

• Helps in factoring expressions or solving them.

• Used in solving algebraic equations quickly without any lengthy calculations.

• Reduces lengthy multiplication into simpler steps

Proof of A² – B² = (A + B)(A – B)

We can prove this using basic multiplication:

(a + b)(a – b)

= a(a – b) + b(a – b)

= a² – ab + ab – b²

= a² – b²

 Hence, (a + b)(a – b) = a² – b²

Note - Just solve step by step to solve any question for any proof-type question.

Examples on A² + B² and A² – B² Formulas

To understand how these formulas work in real problem-solving, go through the following examples. 

Each example shows the application of either the sum of squares or the difference of squares formula with clear, step-by-step solutions for better clarity.

Example 1: Simplify x² – 16

Here, a = x and b = 4
x² – 4² = (x + 4)(x – 4) 

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

Use a² – b² = (a + b)(a – b)

Let A = (3x + 2), B = (3x – 2)

= (A + B)(A – B)

= [(3x + 2) + (3x – 2)] × [(3x + 2) – (3x – 2)]

= (6x) × (4)

= 24x 

Example 3: Calculate 14² + 20² using a² + b²

a = 14, b = 20
a² + b² = 196 + 400 = 596

Example 4: Solve 100² – 8²

Using a² – b² = (a + b)(a – b)
a = 100, b = 8

= (100 + 8)(100 – 8)

= 108 × 92

= 9936 

Relation Between A² + B² and A² – B² Formulas

Both formulas relate to operations on squares, but differ in purpose:

One is used to simplify, the other to factor.

Applications of A² + B² and A² – B² Formulas

The formula a² + b² = (a + b)² – 2ab is widely used in both basic and advanced mathematics, as well as in real-life problem solving. So solve them properly and ask for help from your teachers and parents. Do not feel shy.

1. Simplifying Algebraic Expressions

The sum of squares formula is useful for rewriting expressions in simpler forms, especially during competitive exams. It helps to break larger problems into smaller, more manageable identities.

2. Geometry – Right-Angle Triangle (Pythagoras Theorem)

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
c² = a² + b².
This directly uses the concept of sum of squares.

3. Coordinate Geometry – Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √[(x₂ – x₁)² + (y₂ – y₁)²].
This relies on adding squared differences.

4. Physics – Magnitude of Vectors

In physics, the magnitude of a vector v = ai + bj is:
|v| = √(a² + b²).
This is based on the sum of squares.

5. Trigonometry

Trigonometric identity sin²θ + cos²θ = 1 is an application of sum of squares, widely used in solving trigonometric expressions.

6. Probability and Statistics

Variance uses the sum of squares formula when calculating deviations from the mean:
Variance = Σ(x − mean)² / n.

7. Computer Graphics and Engineering

In graphics programming, calculating distances between objects and determining vector movements often involves squaring and adding numerical values.

Practice Questions (Worksheet)

To help you strengthen your understanding of the A² + B² and A² – B² formulas, try solving the following practice questions. These problems are made for students for their board and competitive exam preparation. Check the step-by-step answers provided below each question:

1. Calculate 15² – 7²
     Answer: (15 + 7)(15 – 7) = 22 × 8 = 176

2. Prove a² + b² = (a + b)² – 2ab
     Answer: (a + b)² = a² + 2ab + b² ⇒ a² + b² = (a + b)² – 2ab

3. Simplify 13² + 6² using a² + b² formula
     Answer: 169 + 36 = 205

4. Evaluate (x + 6)(x – 6)
   Answer: x² – 36 (using a² – b²)

5. Solve 50² – 2²
   Answer: (50 + 2)(50 – 2) = 52 × 48 = 2496