$ 75 imes 75 + 2 imes 75 imes 25 + 25 imes 25 $ is equal to:

Question

$ 75 	imes 75 + 2 	imes 75 	imes 25 + 25 	imes 25 $ is equal to:

Examoogle · Accepted Answer

The correct answer is Option D: 10000. Identify the pattern as $a^2 + 2ab + b^2$, where $a=75$ and $b=25$. Simply calculate $(a + b)^2 = (75 + 25)^2 = 100^2 = 10000$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Identify the pattern as $a^2 + 2ab + b^2$, where $a=75$ and $b=25$. Simply calculate $(a + b)^2 = (75 + 25)^2 = 100^2 = 10000$.

Formula Used: $$(a + b)^2 = a^2 + 2ab + b^2$$

Given: Expression to evaluate: 75 × 75 + 2 × 75 × 25 + 25 × 25

Steps: Identify the algebraic form: Observe the expression $75 	imes 75 + 2 	imes 75 	imes 25 + 25 	imes 25$. Let $a = 75$ and $b = 25$. The expression takes the form $a^2 + 2ab + b^2$.. Apply algebraic identity: Apply the identity $(a + b)^2 = a^2 + 2ab + b^2$ to simplify the expression.. Final calculation: Perform the addition inside the bracket, then square the result.

Answer: D is correct because the expression simplifies to $(75 + 25)^2 = 100^2 = 10000$.

Exam Tip: This identity is fundamental in arithmetic to avoid large multiplications and is frequently tested in banking and railway entrance exams to evaluate complex numerical patterns quickly.

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