Root transformation: If α and β are the roots of 4x^2 − 19x + 12 = 0, find the quadratic equation whose roots are 1/α and 1/β.

Introduction / Context: This tests the reciprocal-root transformation. Given a quadratic with roots α and β, we are to construct the equation for roots 1/α and 1/β without explicitly finding α and β.

Given Data / Assumptions:

• Original equation: 4x^2 − 19x + 12 = 0 with roots α, β.
• We want the equation with roots 1/α and 1/β.
• Coefficients are real and the quadratic is non-degenerate.

Concept / Approach: For ax^2 + bx + c = 0, sum α + β = −b/a and product αβ = c/a. Then for reciprocals: (1/α) + (1/β) = (α + β)/(αβ) and (1/α)(1/β) = 1/(αβ). The monic form is x^2 − [(α + β)/(αβ)]x + [1/(αβ)] = 0, which can be scaled to clear denominators.

Step-by-Step Solution:

Verification / Alternative check: Direct transformation rule: cx^2 + bx + a = 0 for reciprocals gives 12x^2 − 19x + 4 = 0, consistent.

Why Other Options Are Wrong: Signs or placements of coefficients are incorrect; only option with pattern c, b, a works.

Common Pitfalls: Forgetting to divide by product αβ when summing reciprocals; not clearing denominators cleanly.

Final Answer: 12x^2 − 19x + 4 = 0