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
- 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, \(16^2 - 9^2 = 175\).
- 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
- 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, \((x+5)^2 - (x-5)^2 = 20x\).
- 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
- MathWorld - Wolfram. "Difference of Squares." Accessed July 10, 2024. http://mathworld.wolfram.com/DifferenceofSquares.html
- Khan Academy. "Difference of Squares." Accessed July 10, 2024. https://www.khanacademy.org/math