S = rac(a + b)^2 + (a - b)^2(a - b)(a + b) = rac2a^2 + 2b^2a^2 - b^2. - Decision Point
Simplifying the Expression: $ S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} $
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:
$$
S = rac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = rac{2a^2 + 2b^2}{a^2 - b^2}
$$
Understanding the Context
Step-by-Step Simplification
Step 1: Expand the Numerator
Start with the numerator:
$$
(a + b)^2 + (a - b)^2
$$
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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:
$$
(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2
$$
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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.
