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

February 22, 2026

$10^2 = 58 + 2ab \implies 100 = 58 + 2ab \implies 2ab = 42 \implies ab = 21$

Unlocking the Mystery of 10² = 58 + 2ab: Solving for ab = 21

Mathematics is full of elegant equations that reveal deeper truths through logical manipulation. One such intriguing identity is:

10² = 58 + 2ab

While at first glance the equation may seem cryptic, breaking it down step-by-step unlocks a concise solution with value far beyond simple arithmetic. Let’s explore this algebraic mystery and discover how ab = 21 emerges naturally from the algebra.

The Original Equation: A Gateway to Insight

Start with the foundational statement: 10² = 58 + 2ab

We know 10² = 100, so substitute: 100 = 58 + 2ab

This transformation replaces variables with concrete numbers, simplifying the expression for immediate solving.

Step-by-Step Solving to ab = 21

To solve for ab, isolate the term containing the product:

Step 1: Subtract 58 from both sides 100 – 58 = 2ab → 42 = 2ab

Step 2: Divide both sides by 2 42 ÷ 2 = ab → ab = 21

Why This Equation Matters

On the surface, this equation resembles simple algebra, but it highlights how changing or interpreting constants can lead to meaningful results. The choice of 10² (a perfect square) ties the identity to number theory, while 2ab suggests a symmetric product often linked in geometric constructions—such as area problems involving two unknowns, a and b.

Real-World Applications

Equations of this form appear in:

Algebraic word problems where the product of two variables (e.g., dimensions, rates) contributes to a fixed total.
Geometry involving rectangles or areas where one side (10) relates to a known dimension, and the product ab = 21 represents an unknown pairing summing idea with product.
Cryptographic puzzles and educational exercises designed to teach factoring and transformation.

Summary

The equation 10² = 58 + 2ab → 100 = 58 + 2ab → 2ab = 42 → ab = 21 is more than a calculation. It demonstrates how algebraic manipulation bridges numbers and logic to reveal solutions efficiently. Whether for math learners, educators, or problem solvers, understanding such relationships strengthens foundational skills and exposes the beauty in structured reasoning.

Try It Yourself! Can you explore variations with different constants? How do changes in 10², the 58, or 2ab affect the outcome? Such experimentation deepens mathematical intuition and fluency.

By decoding 10² = 58 + 2ab, we uncover not just a number, but a door into systematic problem solving—one predictable step at a time. Explore, verify, and enjoy the power of algebra!