Find the roots of the quadratic equation x^2 − 25x + 156 = 0 and identify the correct ordered pair of solutions.

Introduction / Context:
This problem is a standard quadratic equation solving exercise. It checks whether you can factorise a quadratic expression of the form x^2 − 25x + 156 and correctly identify its two real roots. Such equations appear frequently in algebra and aptitude tests.

Given Data / Assumptions:

• The equation is x^2 − 25x + 156 = 0.
• We look for real roots.
• We can attempt factorisation into (x − p)(x − q) = 0.

Concept / Approach:
Main ideas:

• For x^2 − 25x + 156, find two numbers whose product is 156 and whose sum is 25.
• Use these numbers to factor the quadratic expression.
• Set each factor equal to zero to find the roots.

Step-by-Step Solution:
We seek integers p and q such that p + q = 25 and pq = 156. Try factor pairs of 156: 12 × 13 = 156 and 12 + 13 = 25. So p = 12 and q = 13 satisfy the conditions. Write the quadratic as x^2 − 25x + 156 = (x − 12)(x − 13). Set each factor equal to zero: x − 12 = 0 or x − 13 = 0. Thus x = 12 or x = 13.

Verification / Alternative check:
Substitute x = 12 into the original equation: 12^2 − 25 * 12 + 156 = 144 − 300 + 156 = 0. For x = 13: 169 − 325 + 156 = 0. Both values satisfy the equation, confirming that they are indeed roots.

Why Other Options Are Wrong:
Option b: 25 and 1 have product 25, not 156, so they cannot be the roots of this quadratic.
Option c: 9 and 16 have product 144 and sum 25, but the constant term in the equation is 156, not 144.
Option d: 31 and 6 have product 186 and do not even sum to 25.
Option e: 8 and 17 have product 136, which again is not equal to 156.

Common Pitfalls:
A frequent mistake is to miscalculate factor pairs or to rely on guessing without checking both the sum and product conditions. Another error is to factor incorrectly and then not verify the roots. Systematically listing factor pairs and checking them, or using the quadratic formula as a backup, helps avoid issues.

Final Answer:
The roots of the quadratic equation are 12 and 13.