Difficulty: Easy
Correct Answer: 5
Explanation:
Introduction / Context:
This question involves basic algebraic expansion of two binomials and then extracting coefficients to evaluate a simple expression in terms of those coefficients. Such problems are very common in algebra sections of aptitude tests.
Given Data / Assumptions:
Concept / Approach:
We first expand the product using the distributive property or the FOIL method: multiply each term in the first bracket by each term in the second bracket. Once we have the standard quadratic ax² + kx + n, we read off the coefficients a, k and constant n and substitute them into the expression a - n + k.
Step-by-Step Solution:
Step 1: Expand (3x + 2)(2x - 5).Step 2: Multiply 3x by 2x to get 6x².Step 3: Multiply 3x by -5 to get -15x.Step 4: Multiply 2 by 2x to get 4x.Step 5: Multiply 2 by -5 to get -10.Step 6: Add like terms: 6x² - 15x + 4x - 10 = 6x² - 11x - 10.Step 7: Compare with ax² + kx + n. So a = 6, k = -11 and n = -10.Step 8: Compute a - n + k = 6 - (-10) + (-11) = 6 + 10 - 11 = 5.
Verification / Alternative check:
You can quickly re expand the expression mentally to confirm the coefficients: the x² term comes only from 3x * 2x, the x term comes from combining 3x * -5 and 2 * 2x, and the constant term comes from 2 * -5.Recomputing 6 - (-10) + (-11) as 6 + 10 - 11 confirms 5.
Why Other Options Are Wrong:
4, 3 and 1 result from arithmetic mistakes such as forgetting one of the negative signs or incorrectly adding the coefficients.Any value other than 5 indicates an error in either the expansion step or in evaluating a - n + k.
Common Pitfalls:
Sign errors are very common when dealing with negative numbers, especially in the middle term and constant term.Some students forget to combine the two x terms (-15x and 4x) correctly.Carefully writing each product and then simplifying systematically helps avoid these mistakes.
Final Answer:
The value of a - n + k is 5.
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