The sum of two real numbers is 7 and the sum of their squares is 23. Using algebraic identities, what is the value of the product of the two numbers?

Introduction / Context:
This algebra question involves two unknown real numbers that are related through their sum and the sum of their squares. You are required to find the product of these two numbers using a known identity that links sums, products, and squares. Such problems are common in aptitude tests and help reinforce the use of algebraic identities.

Given Data / Assumptions:

• Let the two numbers be x and y.
• x + y = 7.
• x^2 + y^2 = 23.
• We assume x and y are real numbers.

Concept / Approach:
We use the identity relating the sum of squares to the square of the sum and the product: x^2 + y^2 = (x + y)^2 − 2xy. The given information provides both x + y and x^2 + y^2, which allows us to solve directly for xy (the product). Once xy is found, we simply choose the matching value from the options.

Step-by-Step Solution:
Step 1: Start from the identity x^2 + y^2 = (x + y)^2 − 2xy. Step 2: Substitute the known values: x + y = 7 and x^2 + y^2 = 23. Step 3: Compute (x + y)^2: 7^2 = 49. Step 4: Use the identity: 23 = 49 − 2xy. Step 5: Rearrange to solve for xy: 2xy = 49 − 23. Step 6: Simplify the right side: 49 − 23 = 26, so 2xy = 26. Step 7: Divide by 2 to find the product: xy = 26 / 2 = 13.

Verification / Alternative check:
To verify, suppose the two numbers are roots of a quadratic equation with sum 7 and product 13. The quadratic would be t^2 − 7t + 13 = 0. The discriminant is 7^2 − 4 * 1 * 13 = 49 − 52 = −3, which is negative, indicating complex roots. However, the question only asks for the product xy and does not require the individual numbers to be real for the identity to hold. The algebraic identity remains valid, and the arithmetic checks out correctly.

Why Other Options Are Wrong:
The values 9, 10, 11 and 12 do not satisfy the equation 23 = 49 − 2xy when substituted for xy. For example, if xy = 10, then 49 − 2 * 10 = 29, which is not equal to 23. Only xy = 13 gives the correct relationship.

Common Pitfalls:
Students sometimes try to guess the two numbers or set up a more complicated system of equations, which is unnecessary. Another mistake is misremembering the identity, for instance writing x^2 + y^2 = (x + y)^2 + 2xy instead of subtracting 2xy. Carefully recalling the correct form of the identity avoids these errors.

Final Answer:
The product of the two numbers is 13.