In this aptitude (simplification and quadratic equations) question, you are given that the two roots of a quadratic equation are x = 1/7 and x = -1/8. Using the factor form (x - r1)(x - r2) = 0 and clearing denominators, determine which option represents the correct factorised equation.

Introduction / Context:
This question tests understanding of how to reconstruct a quadratic equation from its roots. Rather than starting from a polynomial and finding its zeros, we are given the roots and must build the factor form of the equation. This is a standard skill in algebra and is useful for many aptitude and competitive exam problems.

Given Data / Assumptions:
The roots are given as x = 1/7 and x = -1/8.
We need the corresponding factor form of the quadratic equation in x.
The equation is assumed to be set equal to zero, that is, of the form something equals zero.

Concept / Approach:
If r1 and r2 are roots of a quadratic, then the factors are (x - r1) and (x - r2). Their product equals zero gives the quadratic. When the roots are fractions, we often multiply both sides by the denominators to eliminate fractions and obtain a neat factorised form with integer coefficients. We then match the resulting factors with the options provided.

Step-by-Step Solution:
Let r1 = 1/7 and r2 = -1/8. The basic factor form is (x - r1)(x - r2) = 0. So the equation is (x - 1/7)(x + 1/8) = 0. Now clear denominators by multiplying each factor by the denominator of its corresponding root. For x - 1/7, multiply by 7 to get 7x - 1. For x + 1/8, multiply by 8 to get 8x + 1. Thus the factorised equation with integer coefficients is (7x - 1)(8x + 1) = 0.

Verification / Alternative check:
We can verify by checking that each given root satisfies the factorised equation. For x = 1/7, the factor 7x - 1 becomes 7 * (1/7) - 1 = 1 - 1 = 0, so the product is zero and the root is valid. For x = -1/8, the factor 8x + 1 becomes 8 * (-1/8) + 1 = -1 + 1 = 0, again giving zero. Therefore the factor pair (7x - 1) and (8x + 1) correctly corresponds to the two given roots.

Why Other Options Are Wrong:
Option (7x - 1)(8x - 1) = 0 would have roots 1/7 and 1/8, both positive, which does not match the negative second root. Option (7x + 1)(8x - 1) = 0 would give roots -1/7 and 1/8. Option (7x + 1)(8x + 1) = 0 would give roots -1/7 and -1/8. The final option (56x - 1)(x + 1) = 0 has roots 1/56 and -1, again inconsistent with the given pair. Only the chosen factorisation produces exactly 1/7 and -1/8 as roots.

Common Pitfalls:
Learners often forget that the factor is x minus the root, not x plus the root, leading to sign errors. Another mistake is to ignore denominators and try to write factors directly with fractional constants, making comparison with the options difficult. Multiplying each factor by the denominator removes fractions and leads to cleaner expressions that are easier to match.

Final Answer:
The correct factor form of the quadratic equation with roots 1/7 and -1/8 is (7x - 1)(8x + 1) = 0.