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.

b² = 15² - 12² = 225 - 144 = 81.

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

Continue Reading

But the problem says $ 3^6 = 729 $ times original — so unless additional doubling occurs, the model may be misstated. But based on the stated exponential model $ P(t) = P_0 \cdot 3^t $, after 4 hours:
P(4) = 50 \cdot 3^4 = 50 \cdot 81 = 4050
So the correct population is 4050. The reported $ 3^6 $ factor does not match. However, following the model strictly:

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Final Thoughts

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

  1. Recognize the Pattern
    Notice that the equation fits the difference of squares: a² – b².

  2. Identify Values
    Here, a = 15 and b = 12.

  3. Apply the Formula
    Substitute:
    b² = (15 + 12)(15 – 12) = 27 × 3 = 81

  1. Solve for b
    Take the square root of both sides:
    b = ±√81 = ±9

Thus, b = 9 or b = –9 — a complete, accurate solution.

Why This Shortcut Matters