Solve for $ ab $ Given $ a + b = 10 $ and $ a^2 + b^2 = 58 $

Understanding the relationship between sum and squares is key in algebra—especially when solving equations involving two variables. In this article, we explore how to find the product $ ab $ when given $ a + b = 10 $ and $ a^2 + b^2 = 58 $. This seemingly simple problem reveals powerful techniques used in solving quadratic relationships and contributes to real-world applications in physics, engineering, and data analysis.

The Equation Setup

Understanding the Context

We are given:
$$
a + b = 10
$$
$$
a^2 + b^2 = 58
$$
Our goal is to find the value of $ ab $.

Use the Identity for $ (a + b)^2 $

Recall the fundamental algebraic identity:
$$
(a + b)^2 = a^2 + 2ab + b^2
$$
We can rearrange this to solve for $ ab $:
$$
(a + b)^2 = a^2 + b^2 + 2ab
$$

Substitute Known Values

Sabemos $a + b = 10$ y $a^2 + b^2 = 58$. Para encontrar $ab$, usamos:

Key Insights

Plug in the given values:
$$
10^2 = 58 + 2ab
$$
$$
100 = 58 + 2ab
$$

Solve for $ 2ab $

Subtract 58 from both sides:
$$
100 - 58 = 2ab
$$
$$
42 = 2ab
$$

Find $ ab $

Divide both sides by 2:
$$
ab = rac{42}{2} = 21
$$

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Final Thoughts

Conclusion

The product of $ a $ and $ b $ is:
$$
oxed{21}
$$

This elegant solution shows how algebraic identities simplify complex problems. Whether you're solving for variables in math contests, deriving formulas in physics, or modeling relationships in data science, understanding how sums and squares connect to products is invaluable. Remember: sometimes, all you need to unlock a mystery is one key identity.

By mastering such techniques, you enhance your problem-solving toolkit and improve your grasp of algebra’s core principles—essential for academic success and practical applications.
Keywords: $ a + b = 10 $, $ a^2 + b^2 = 58 $, find $ ab $, algebraic identity, solve for product, mathematical technique, algebra practice.