Difficulty: Easy
Correct Answer: 169
Explanation:
Introduction / Context:
This algebra question checks your understanding of how to use identities involving sums and differences of numbers. Instead of solving directly for a and b, you can exploit the relationship between (a − b)^2, ab, and a^2 + b^2 to quickly compute the desired value.
Given Data / Assumptions:
Concept / Approach:
The key identity here is (a − b)^2 = a^2 + b^2 − 2ab. We already know both a − b and ab, so we can rearrange this identity to express a^2 + b^2 in terms of them. This avoids solving a full quadratic to find a and b individually.
Step-by-Step Solution:
Step 1: Write the identity (a − b)^2 = a^2 + b^2 − 2ab.Step 2: Substitute the known value a − b = 11 into the identity. Then (a − b)^2 = 11^2 = 121.Step 3: Replace (a − b)^2 with 121 and ab with 24 in the identity, giving 121 = a^2 + b^2 − 2 * 24.Step 4: Compute 2 * 24 = 48, so we have 121 = a^2 + b^2 − 48.Step 5: Add 48 to both sides to isolate a^2 + b^2: a^2 + b^2 = 121 + 48 = 169.
Verification / Alternative check:
As an optional check, solve for a and b explicitly. Consider the quadratic t^2 − (a + b)t + ab = 0. We know a − b = 11 and ab = 24. If you also find a + b from (a − b)^2 and (a + b)^2 relations, you can compute actual values that satisfy the conditions and then verify that a^2 + b^2 = 169. However, this is not necessary because the identity already gives a direct answer.
Why Other Options Are Wrong:
- 37 is obtained if someone mistakenly computes 121 − 2 * 42 or mixes up numbers, which is incorrect.
- 73 and 48 do not follow from the identity and represent incorrect algebraic manipulation.
- 121 is simply (a − b)^2, not a^2 + b^2.
Common Pitfalls:
A common error is to forget the negative sign in front of 2ab or misapply the identity. Some students also confuse (a + b)^2 and (a − b)^2. Always write the identity carefully and substitute step by step to avoid arithmetic mistakes.
Final Answer:
The value of a^2 + b^2 is 169.
Discussion & Comments