Difficulty: Easy
Correct Answer: 9x^2 - 30x^2*y + 25x^2*y^2
Explanation:
Introduction / Context:This question tests the standard algebra identity for squaring a binomial: (a - b)^2 = a^2 - 2ab + b^2. Here, the terms are not simple numbers; they are algebraic expressions (3x and 5xy). The only real task is to apply the identity carefully and multiply coefficients and variables correctly.
Given Data / Assumptions:
Concept / Approach:Compute a^2, compute 2ab, compute b^2, then combine with the correct signs: a^2 - 2ab + b^2.
Step-by-Step Solution:
Step 1: Let a = 3x and b = 5xy. Step 2: Compute a^2: (3x)^2 = 9x^2. Step 3: Compute 2ab: 2*(3x)*(5xy) = 2*15*x*x*y = 30x^2*y. Step 4: Compute b^2: (5xy)^2 = 25*x^2*y^2 = 25x^2*y^2. Step 5: Apply formula: (3x - 5xy)^2 = a^2 - 2ab + b^2 = 9x^2 - 30x^2*y + 25x^2*y^2.Verification / Alternative check:Quick check using distributive multiplication: (3x - 5xy)(3x - 5xy) gives 9x^2 from 3x*3x, -15x^2 y twice from cross terms totaling -30x^2 y, and +25x^2 y^2 from (-5xy)(-5xy). Same result.
Why Other Options Are Wrong:
-15 term options: they forget there are two cross terms. Swapped coefficients: they confuse (3x)^2 with (5xy)^2. +30 option: sign error; cross term must be negative in (a - b)^2.Common Pitfalls:Forgetting the factor 2 in -2ab, or mishandling x*x as x^2, or losing the negative sign in the cross term.
Final Answer:9x^2 - 30x^2*y + 25x^2*y^2
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