b² = 15² - 12² = 225 - 144 = 81. - Decision Point
Title: Solve the Powerful Difference of Squares: b² = 15² – 12² – A Simple Math Shortcut That Solves to 81
Title: Solve the Powerful Difference of Squares: b² = 15² – 12² – A Simple Math Shortcut That Solves to 81
Subtitle: Discover how b² = 15² – 12² = 81 using factorization, difference of squares, and real-world applications
Understanding the Context
Mathematics isn’t just about numbers—it’s about elegant patterns and clever shortcuts. One powerful algebraic identity you’ll rarely hear about is the difference of squares, and a classic example is:
> b² = 15² – 12² = 225 – 144 = 81
This equation simplifies beautifully to b² = 81, revealing that b = ±9 — a quick, insightful solution that many students overlook. In this article, we’ll explore how to solve this using the difference of squares formula, why it’s useful, and how understanding this concept can strengthen your math skills.
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Key Insights
What Is the Difference of Squares Formula?
The difference of squares formula is:
a² – b² = (a + b)(a – b)
This identity allows you to factor quadratic expressions quickly and solve equations like b² = 15² – 12² without resorting to basic multiplication or trial and error.
In our example:
- a = 15 (since 15² is the larger square)
- b = 12 (whose square is subtracted)
Applying the formula:
15² – 12² = (15 + 12)(15 – 12) = 27 × 3 = 81
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So instead of calculating 15² = 225 and 12² = 144 separately, then subtracting, we factor directly — saving time and mental effort.
How to Solve b² = 15² – 12² Step-by-Step
-
Recognize the Pattern
Notice that the equation fits the difference of squares: a² – b². -
Identify Values
Here, a = 15 and b = 12. -
Apply the Formula
Substitute:
b² = (15 + 12)(15 – 12) = 27 × 3 = 81
- Solve for b
Take the square root of both sides:
b = ±√81 = ±9
Thus, b = 9 or b = –9 — a complete, accurate solution.
