S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = rac{2a^2 + 2b^2}{a^2 - b^2}.

February 22, 2026

$S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = rac{2a^2 + 2b^2}{a^2 - b^2}.$

Simplifying the Expression: $ S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} $

Mathematics is full of elegant simplifications, and one particularly insightful expression involves simplifying a compound fraction to reveal its underlying structure. Here, we explore the simplification of:

$$ S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = rac{2a^2 + 2b^2}{a^2 - b^2} $$

Step-by-Step Simplification

Step 1: Expand the Numerator

Start with the numerator: $$ (a + b)^2 + (a - b)^2 $$

Using the identity $(x + y)^2 = x^2 + 2xy + y^2$, expand both squares:

$$ (a + b)^2 = a^2 + 2ab + b^2 $$ $$ (a - b)^2 = a^2 - 2ab + b^2 $$

Now add them:

$$ (a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2 $$

Step 2: Simplify the Denominator

The denominator is: $$ (a - b)(a + b) $$

This is a difference of squares: $$ (a - b)(a + b) = a^2 - b^2 $$

Step 3: Rewrite $S$ with the Simplified Parts

Now substitute both simplified forms back into $S$:

$$ S = rac{2a^2 + 2b^2}{a^2 - b^2} $$

This matches the given simplified form.

Final Result

$$ oxed{S = rac{2a^2 + 2b^2}{a^2 - b^2}} $$

Why This Simplification Matters

Expressing $S$ in this compact form highlights its dependence on sum-of-squares in the numerator and a difference-of-squares in the denominator. This connection is valuable in algebra, trigonometry, and even physics, where such expressions model phenomena involving energy, symmetry, or relative motion.

Key Takeaways:

Expand expressions carefully before combining terms.
Recognize algebraic identities: $(a + b)^2$, $(a - b)^2$, and difference of squares.
Simplify numerator and denominator separately, then recombine.

Takeaway for Students and Professionals

Understanding how to simplify complex rational expressions increases fluency in algebra and opens doors to tackling higher-level problems. Whether graphing functions or solving real-world equations, the form $ rac{2a^2 + 2b^2}{a^2 - b^2} $ is cleaner, more interpretable, and easier to manipulate than the original expanded or factored version.