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

  1. Compute 10²
    First, calculate the square of 10:
    10² = 100

  2. Rewrite the inequality
    Substitute the value into the equation:
    100 – 2ab ≤ 58

= 10^2 - 2ab \implies 58 = 100 - 2ab \implies 2ab = 42 \implies ab = 21

Key Insights

  1. Isolate the term with ‘ab’
    Subtract 100 from both sides to simplify:
    –2ab ≤ 58 – 100
    –2ab ≤ –42

  2. Eliminate the negative sign
    Multiply both sides by –1 (remember: multiplying by a negative reverses the inequality):
    2ab ≥ 42

  3. Solve for ab
    Divide both sides by 2:
    ab ≥ 21

Key Insight: Finding the Exact Value

Continue Reading

\(f''(1) = 18(1) - 4 = 14\) (positive, indicating a local minimum if \(f'(1) = 0\))
Since \(f'(1)
eq 0\), \(x = 1\) is neither a maximum nor a minimum.

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Final Thoughts

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.