If a + b = 10 and the product ab = 21 for real numbers a and b, then what is the value of (a − b)^2?

Introduction / Context:
This algebra problem works with simple symmetric expressions in two variables. Instead of solving individually for a and b, we use identities relating (a − b)^2 to a + b and ab, which are already given.

Given Data / Assumptions:

• a and b are real numbers.
• a + b = 10.
• ab = 21.
• We must find (a − b)^2.

Concept / Approach:
We use the identity (a − b)^2 = (a + b)^2 − 4ab. Because both a + b and ab are known, we can compute (a − b)^2 directly without solving a quadratic equation for a and b. This is a standard and efficient technique.

Step-by-Step Solution:
Step 1: Recall the identity (a − b)^2 = (a + b)^2 − 4ab. Step 2: Substitute a + b = 10 to get (a + b)^2 = 10^2 = 100. Step 3: Substitute ab = 21, so 4ab = 4 × 21 = 84. Step 4: Compute (a − b)^2 = 100 − 84 = 16. Step 5: Therefore the value of (a − b)^2 is 16.

Verification / Alternative check:
We can also find a and b explicitly. Consider the quadratic t^2 − (a + b)t + ab = 0, which becomes t^2 − 10t + 21 = 0. This factors as (t − 3)(t − 7) = 0, so a and b are 3 and 7 in some order. Then a − b is either 3 − 7 = −4 or 7 − 3 = 4, and in both cases (a − b)^2 = 16, confirming our earlier result.

Why Other Options Are Wrong:
Options A (15), C (17), and D (18) are close but incorrect and usually result from arithmetic slips such as computing 4ab incorrectly or miscalculating (a + b)^2. Option E (20) has no basis in the identity and is simply a distractor for those who do not apply the correct formula.

Common Pitfalls:
Some learners forget the difference between (a − b)^2 and a^2 − b^2. Using the wrong identity leads to errors. Others attempt to solve for a and b with the quadratic formula unnecessarily, which can create numerical mistakes. Using the direct identity is faster and more reliable in test situations.

Final Answer:

The value of (a − b)^2 is 16.