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The a² + b² formula and its counterpart a² – b² constitute fundamental algebraic expressions that hold significant importance in mathematics. These formulas are widely taught and applied starting from the 10th grade and continue to play a crucial role in various educational settings, competitive exams, and real-world problem-solving scenarios.

The a² + b² formula represents the sum of squares of two variables a and b, while a² – b² signifies the difference of squares. These expressions find applications in geometry, trigonometry, physics, engineering, statistics, and even complex number calculations. Mastering these basic algebraic formulas equips individuals with powerful tools to tackle mathematical challenges easily and efficiently.

## a^{2} b^{2} Formula a²+b²

Let us consider that a and b are two mathematical variables that 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 expression a² + b² represents a binomial algebraic equation with ‘a’ and ‘b’ as variables denoting two terms in algebra. When the squares of the respective terms are added, it will be written as a2 + b2.

**Derivation of **a²+b²

As we all know, (a+b)^{2} = a^{2}+b^{2}+2ab

a^{2}+b^{2} = (a+b)^{2} – 2ab

Also, we know that (a-b)v2 = a^{2}+b^{2}-2ab

a^{2}+b^{2} = (a-b)^{2} + 2ab

So, formulas related to a²+b² derived above are concluded below as,

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

This fundamental formula a² + b² finds wide applications in various disciplines. Geometrically, it corresponds to the Pythagorean theorem, which is crucial in proving geometric relationships and trigonometric identities. a² + b² calculates magnitudes, distances, and forces in physics and engineering, essential in mechanics, electricity, and waves. Statistics determines variances and standard deviations, providing insights into data dispersion. The formula is also pertinent in working with complex numbers, employed in modulus and absolute value representation and linked to Euler’s formula. Its versatility and significance make it a cornerstone in mathematics and practical applications.

### a^{2} b^{2} Formula a²-b²

The expression a² – b² represents a binomial algebraic equation, where ‘a’ and ‘b’ are mathematical variables signifying two terms in algebra.

### Derivation of a^{2} b^{2}

The a² – b² formula can be derived geometrically using subtracting the area of a small square from a larger square. This geometric representation helps us understand how the difference of squares can be factored into the product of (a + b) and (a – b). The area of the subtracted shape, a square of side b, is equal to a² – b². By rearranging the shape and forming a rectangle with sides (a + b) and (a – b), we can observe that the area of the rectangle is also equal to a² – b².

Thus, we arrive at the factorisation a² – b² = (a + b)(a – b).

This fundamental formula holds significant importance across various mathematical disciplines and practical applications. Geometrically, it relates to the difference of squares, facilitating polynomial factorisation and special geometric shapes analysis. In trigonometry, a² – b² is instrumental in deriving identities and solving complex equations involving sines, cosines, and tangents. The formula finds applications in mechanics, optics, and electrical circuits in physics and engineering, contributing to the derivation of crucial equations. Understanding and applying a² – b² empower professionals to tackle intricate problems efficiently, making notable advancements in their respective fields.

### a^{2} b^{2}** Formula: Examples**

**Question:** using the sum of squares formula, calculate the value of (2)² + ()03².

Solution:

Given that the value of a = 2, b = 3

By using the a² + b² Formula,

a² + b² =(a +b)² – 2ab

= (2 + 3)² – 2 (2)(3)

= 5² – 2 (6)

= 25 – 12

= 13

Therefore, a² + b² = 13

**Question:** Using the formula of the square, find the value of the given expression 2² – 3².

Solution:

Given that the value of a = 2, b = 3

By using the formula of the square,

a² + b² = (a + b)² − 2ab

= (2 + 3)(2 – 3)

= (5)(-1)

= -5

Therefore, a² – b² = -5

**Related Links**

(a + b) Whole Cube Formula | Trigonometry Formulas |

Standard deviation formula | Maths Formulas |

## Frequently Asked Questions (FAQs) on a^{2} b^{2 }Formula

### What does the a² + b² formula represent?

The a² + b² formula represents the sum of squares of two variables 'a' and 'b' in algebraic expressions.

### How is the a² + b² formula derived geometrically?

The a² + b² formula can be derived geometrically by adding the square of two algebraic terms corresponding to a square box's area.

### What does the a² - b² formula signify?

The a² - b² formula represents the difference of squares of two variables 'a' and 'b' in algebraic expressions.