Understanding the Difference of Squares Formula With Examples

Introduction

The formula \(a^2 - b^2 = (a - b)(a + b)\) represents the difference of squares, a crucial algebraic identity that simplifies the subtraction of two squares into a product of two factors. This formula is not only foundational in algebra but also serves in simplifying expressions and solving equations across different areas of mathematics.

Numeric Examples

  1. Example 1:

    Calculate \(16^2 - 9^2\).

    Solution:

    • Identify \(a = 16\) and \(b = 9\).
    • Apply the formula: \(16^2 - 9^2 = (16 - 9)(16 + 9)\).
    • Calculate: \(7 \times 25 = 175\).
    • Thus, \(16^2 - 9^2 = 175\).
  2. Example 2:

    Compute \(25^2 - 15^2\).

    Solution:

    • Identify \(a = 25\) and \(b = 15\).
    • Using the formula: \(25^2 - 15^2 = (25 - 15)(25 + 15)\).
    • Perform the calculations: \(10 \times 40 = 400\).
    • Hence, \(25^2 - 15^2 = 400\).

Literal Examples

  1. Example 3:

    Simplify \((x+5)^2 - (x-5)^2\).

    Solution:

    • Apply the formula by substituting \(a = x+5\) and \(b = x-5\): \((x+5)^2 - (x-5)^2 = ((x+5) - (x-5))((x+5) + (x-5))\).
    • Simplify inside the brackets: \(10 \times (2x) = 20x\).
    • Thus, \((x+5)^2 - (x-5)^2 = 20x\).
  2. Example 4:

    Simplify \((3y)^2 - (y)^2\).

    Solution:

    • Identify \(a = 3y\) and \(b = y\).
    • Using the formula: \((3y)^2 - y^2 = (3y - y)(3y + y)\).
    • Calculate: \(2y \times 4y = 8y^2\).
    • Therefore, \((3y)^2 - y^2 = 8y^2\).

References