(a-b)^2=a^2-2ab+b^2 Formula Name

4 min read Jun 16, 2024

The Square of a Difference: Understanding (a - b)² = a² - 2ab + b²

The formula (a - b)² = a² - 2ab + b² is a fundamental algebraic identity that describes the expansion of the square of a difference. This formula is extremely useful in simplifying expressions, solving equations, and performing various algebraic operations.

Understanding the Formula

The formula states that squaring a binomial of the form (a - b) results in the sum of the squares of the individual terms (a² and b²) minus twice the product of the two terms (2ab).

Here's a breakdown:

(a - b)²: This represents the square of the binomial (a - b).
a²: This is the square of the first term (a).
b²: This is the square of the second term (b).
-2ab: This represents twice the product of the first and second terms (a and b).

Proof of the Formula

We can prove the formula using the distributive property of multiplication:

(a - b)² = (a - b)(a - b)

Expanding this product, we get:

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

Therefore, we have successfully proven the formula (a - b)² = a² - 2ab + b².

Applications of the Formula

This formula has widespread applications in various mathematical fields:

Simplifying expressions: It allows us to expand squares of binomials, simplifying complex expressions.
Solving equations: It helps in solving quadratic equations by factoring or completing the square.
Geometry: It can be used to find the area of squares and other geometric figures.
Calculus: It is used in deriving formulas for derivatives and integrals.

Examples

Here are some examples of how to apply the formula:

1. Expanding (x - 3)²:

Using the formula, we get:

(x - 3)² = x² - 2(x)(3) + 3² = x² - 6x + 9

2. Solving the equation (x - 2)² = 9:

Expanding the left side using the formula:

x² - 4x + 4 = 9 x² - 4x - 5 = 0

Factoring the quadratic equation, we get:

(x - 5)(x + 1) = 0

Therefore, the solutions are x = 5 and x = -1.

Conclusion

The formula (a - b)² = a² - 2ab + b² is a fundamental identity that simplifies algebraic expressions and has numerous applications in various mathematical fields. By understanding and applying this formula, we can effectively solve equations, simplify expressions, and gain a deeper understanding of algebraic concepts.