= 10^2 - 2ab \implies 58 = 100 - 2ab \implies 2ab = 42 \implies ab = 21 - Redraw
Understanding the Equation: 10² – 2ab ≤ 58 Leading to ab = 21
Understanding the Equation: 10² – 2ab ≤ 58 Leading to ab = 21
Mathematical expressions often serve as the backbone for problem-solving in algebra, logic puzzles, and real-world applications. One such elegant equation—10² – 2ab ≤ 58—might seem abstract at first glance, but carefully breaking it down reveals a clear path to finding the product ab = 21. In this article, we’ll explore how this inequality unfolds step-by-step and why it’s a classic example of algebraic reasoning.
Understanding the Context
Step-by-Step Breakdown
The inequality begins with a numerical baseline:
10² – 2ab ≤ 58
-
Compute 10²
First, calculate the square of 10:
10² = 100 -
Rewrite the inequality
Substitute the value into the equation:
100 – 2ab ≤ 58
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Key Insights
-
Isolate the term with ‘ab’
Subtract 100 from both sides to simplify:
–2ab ≤ 58 – 100
–2ab ≤ –42 -
Eliminate the negative sign
Multiply both sides by –1 (remember: multiplying by a negative reverses the inequality):
2ab ≥ 42 -
Solve for ab
Divide both sides by 2:
ab ≥ 21
Key Insight: Finding the Exact Value
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So far, we know ab ≥ 21. But what if the original inequality is actually an equality? Consider refined problem contexts, such as optimization or equality-based constraints, where = takes precedence:
10² – 2ab = 58
Following the same steps:
- Start with 100 – 2ab = 58
- Subtract 100: –2ab = –42
- Divide by –2: ab = 21
This confirms that ab = 21 is the precise solution under equality, making it both algebraically sound and practically meaningful.
Why This Equation Matters
Equations like 10² – 2ab = 58 appear frequently in competitive math, physics, and engineering. They model balance—where the total (100) decreases by twice the product of two variables (2ab)—to match a target value (58). Understanding such relationships helps in:
- Optimization: Finding maximum/minimum values under constraints
- Geometry: Relating areas, perimeters, or coefficients in coordinate problems
- Algebraic Proof: Demonstrating equivalences and logical transformations
Practical Takeaways
- Always simplify constants first to reveal underlying structure.
- Be cautious with inequality signs—multiplying/dividing by negatives flips directions, but division by positive numbers preserves them.
- Equality conduces to precise answers; inequalities bound possibilities.
