6^2 = 130 - 2ab \Rightarrow 36 = 130 - 2ab \Rightarrow 2ab = 94 \Rightarrow ab = 47 - Nelissen Grade advocaten
Understanding the Equation: From 6² = 130 – 2ab to ab = 47
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.
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
🔗 Related Articles You Might Like:
📰 A seismologist is analyzing fault line activity and detects aftershocks following a major earthquake. On the first day, 24 aftershocks are recorded. On each subsequent day, the number of aftershocks decreases by half. How many total aftershocks are recorded by the end of the fourth day? 📰 Day 1: 24 📰 Day 2: 24 ÷ 2 = 12Final 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.
