In algebra, two real numbers a and b satisfy a − b = −7 and a^2 + b^2 = 85. Using these conditions, find the exact value of the product ab.

Introduction / Context:
This algebra question focuses on using identities that relate sums and differences of numbers to their squares and products. Instead of asking you to solve directly for a and b, it requires you to manipulate expressions like a^2 + b^2 and a − b to deduce the product ab. Such questions appear frequently in aptitude exams and help develop fluency with algebraic identities and symmetric expressions.

Given Data / Assumptions:

• a and b are real numbers.
• a − b = −7.
• a^2 + b^2 = 85.
• We are asked to find the value of ab.
• No additional restrictions on a and b are given.

Concept / Approach:
The key identity is based on squaring a difference: (a − b)^2 = a^2 + b^2 − 2ab. We already know both a − b and a^2 + b^2, so we can substitute these values into the identity and solve the resulting linear equation for ab. This avoids having to solve a quadratic equation for a and b individually and makes the computation quick and efficient.

Step-by-Step Solution:
1) Start with the identity (a − b)^2 = a^2 + b^2 − 2ab. 2) Substitute the given value a − b = −7. Then (a − b)^2 = (−7)^2 = 49. 3) Substitute a^2 + b^2 = 85 into the identity: 49 = 85 − 2ab. 4) Rearrange to isolate ab: move 85 to the left or 49 to the right. Subtract 85 from both sides to get 49 − 85 = −2ab. 5) Compute 49 − 85 = −36, so −36 = −2ab. 6) Divide both sides by −2: ab = (−36) / (−2) = 18.

Verification / Alternative check:
We can verify the result by constructing a quadratic whose roots are a and b. Let t be a variable with roots a and b, so t^2 − (a + b)t + ab = 0. While a + b is not given directly, we can find it using (a − b)^2 + 4ab = (a + b)^2. We already know (a − b)^2 = 49 and ab = 18, so (a + b)^2 = 49 + 4 * 18 = 49 + 72 = 121, giving a + b = 11 or −11. Using these values, you can solve for a and b explicitly and check that a^2 + b^2 indeed equals 85, which confirms that ab = 18 is consistent.

Why Other Options Are Wrong:
Option b (−18) would require (a − b)^2 to equal 85 + 2 * (−18) = 49, but this sign choice conflicts with the identity when computed carefully. Options c (30), d (−30), and e (44) all give values of a^2 + b^2 that do not match 85 when substituted into the identity. Only ab = 18 makes the identity (a − b)^2 = a^2 + b^2 − 2ab hold with the given numerical values.

Common Pitfalls:
A frequent mistake is to mis remember the identity as a^2 + b^2 = (a − b)^2 + 2ab, which is actually correct but often rearranged incorrectly when solving for ab. Another common issue is incorrect arithmetic when computing 49 − 85. Some learners also forget to keep track of negative signs and may end up with ab = −18 by accident. Careful substitution and step by step rearrangement prevent these errors.

Final Answer:
The product of the two numbers that satisfies the given conditions is 18.