Form an equation with roots α/β and β/α: Given that α and β are roots of 2x^2 − 3x + 1 = 0, form the quadratic equation whose roots are α/β and β/α.

Introduction / Context:Transforming roots often relies on expressing new symmetric sums in terms of the old. If α and β satisfy a known quadratic, we can compute α + β and αβ, then find (α/β) + (β/α) and (α/β)(β/α) to construct the desired equation.Given Data / Assumptions:

• Original equation: 2x^2 − 3x + 1 = 0.
• Hence α + β = 3/2, αβ = 1/2.
• New roots: r1 = α/β and r2 = β/α.

Concept / Approach:Compute sum and product of new roots: r1 + r2 = (α^2 + β^2)/(αβ) and r1*r2 = 1. Then use x^2 − (sum)x + (product) = 0 and clear denominators to get integer coefficients.

Step-by-Step Solution:

Verification / Alternative check:Symmetry implies r1 and r2 are reciprocals; indeed product 1 confirms. Coefficients are integers after clearing denominators.

Why Other Options Are Wrong:

• 2x^2 + 5x + 2 = 0 or 2x^2 − 5x − 2 = 0: Wrong sign patterns for sum and product.
• None of these: Incorrect since 2x^2 − 5x + 2 = 0 fits perfectly.

Common Pitfalls:Miscomputing α^2 + β^2 or forgetting to divide by αβ. Ensure careful fraction arithmetic.

Final Answer: