Find two numbers — Their sum is 15 and the sum of their squares is 113. Identify the pair that satisfies both conditions.

Introduction / Context:
When given a sum and the sum of squares, you can avoid guessing by using identities. Specifically, (a + b)^2 = a^2 + b^2 + 2ab lets you deduce the product ab and then solve a simple quadratic whose roots are the numbers themselves. This is a staple technique in algebra-focused aptitude questions.

Given Data / Assumptions:

• a + b = 15.
• a^2 + b^2 = 113.
• We seek ordered numbers that satisfy both constraints.

Concept / Approach:
Use (a + b)^2 = a^2 + b^2 + 2ab to find ab. Once ab is known, the numbers are roots of t^2 - (a + b)t + ab = 0. Solve and select the pair that matches standard integer solutions.

Step-by-Step Solution:
Compute 2ab from identity: (a + b)^2 - (a^2 + b^2) = 2ab.(15)^2 - 113 = 225 - 113 = 112 → 2ab = 112 → ab = 56.Form quadratic: t^2 - 15t + 56 = 0.Factor: (t - 7)(t - 8) = 0 → t = 7 or 8.Hence, the numbers are 7 and 8.

Verification / Alternative check:
Sum 7 + 8 = 15 and 7^2 + 8^2 = 49 + 64 = 113, satisfying both conditions exactly.

Why Other Options Are Wrong:

• 4,11 / 5,10 / 6,9 / 3,12: Only 7,8 yields both the correct sum and correct sum of squares.

Common Pitfalls:
Miscomputing (a + b)^2; forgetting to divide by 2 when solving for ab; arithmetic mistakes when factoring the quadratic.

Final Answer:
7, 8