Understanding the Equation: From 6² = 130 – 2ab to ab = 47

Mathematics is full of elegant transformations—where simple equations reveal deeper truths. One such example involves the expression 6² = 130 – 2ab, leading step-by-step to the elegant result ab = 47. This method showcases how algebraic manipulation unlocks insights in number theory and problem-solving. Let’s explore the derivation and its implications.

Understanding the Context

The Equation: 6² = 130 – 2ab

We begin with a carefully structured equation:
6² = 130 – 2ab
We know 6² = 36, so substitute to get:
36 = 130 – 2ab

This equation invites us to isolate the product term 2ab, a common algebraic technique used across math problems—from geometry to number theory.

6^2 = 130 - 2ab \Rightarrow 36 = 130 - 2ab \Rightarrow 2ab = 94 \Rightarrow ab = 47

Key Insights

Step 1: Rearranging to Isolate the Product

Subtract 130 from both sides:
36 – 130 = –2ab
–94 = –2ab
Multiply both sides by –1 to simplify:
94 = 2ab

Step 2: Solving for ab

Now divide both sides by 2:
2ab = 94 ⇒ ab = 47

Continue Reading

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Question: An ornithologist tracks a bird’s flight path, which follows a parabolic trajectory modeled by $ y = -x^2 + 6x + 1 $. Determine the maximum height reached by the bird.
Solution: The maximum height of a parabola $ y = ax^2 + bx + c $ occurs at $ x = -\frac{b}{2a} $. Here, $ a = -1 $, $ b = 6 $, so $ x = -\frac{6}{2(-1)} = 3 $. Substituting $ x = 3 $ into the equation:

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Final Thoughts

This final step reveals a simple yet powerful insight: the product of two variables, a and b, equals 47, given the constraint based on a = 6 and the equation 6² = 130 – 2ab.

Why This Matters: Algebraic Insight and Problem Solving

This derivation highlights key algebraic principles:

  • Substitution: Replacing known values to simplify equations
  • Isolation: Strategically rearranging terms to solve for unknown variables
  • Simplification: Reducing expressions step-by-step for clarity

Such reasoning is essential in solving quadratic expressions, factoring, and balancing equations—skills foundational in algebra, calculus, and advanced math applications.

Real-World Context: Applications of ab = 47

In practical scenarios, knowing that ab = 47 under certain conditions helps in optimization problems, combinatorics, or factoring polynomials. For example:

  • A rectangle with one side fixed at 6 and area derived from ab = 47 implies the other side is 47/6.
  • In algebra, this might appear in expressions like (x – a)(x – b), where a = 6 and ab = 47 leads to a recognizable quadratic pattern.

Conclusion