Let the roots of the quadratic equation 2x^2 + 5x + 3 = 0 be r1 and r2. Which of the following quadratic equations has roots equal to the reciprocals 1/r1 and 1/r2?

Introduction / Context:
This question examines your understanding of how the coefficients of a quadratic equation relate to its roots, and in particular, how to find the quadratic equation whose roots are the reciprocals of the roots of a given quadratic. It uses Vieta relations that connect sums and products of roots with the coefficients of the equation.

Given Data / Assumptions:

• Original equation: 2x^2 + 5x + 3 = 0.
• Roots of this equation are r1 and r2 (not explicitly given).
• We need an equation whose roots are 1/r1 and 1/r2.
• Options propose candidate quadratic equations in standard form.

Concept / Approach:
For a quadratic ax^2 + bx + c = 0 with roots r1 and r2, Vieta relations state that r1 + r2 = −b/a and r1r2 = c/a. If we define new roots s1 = 1/r1 and s2 = 1/r2, then their sum and product are: s1 + s2 = (r1 + r2)/(r1r2) and s1s2 = 1/(r1r2). Once we know these, we can form the new quadratic equation as x^2 − (s1 + s2)x + s1s2 = 0 and clear denominators to match one of the options.

Step-by-Step Solution:
For 2x^2 + 5x + 3 = 0, we have a = 2, b = 5, c = 3.Sum of roots r1 + r2 = −b/a = −5/2.Product of roots r1r2 = c/a = 3/2.Now, s1 = 1/r1 and s2 = 1/r2. Then s1 + s2 = (r1 + r2)/(r1r2) = (−5/2) / (3/2) = −5/3.Also, s1s2 = 1/(r1r2) = 1 / (3/2) = 2/3.So the required quadratic with roots 1/r1 and 1/r2 is x^2 − (s1 + s2)x + s1s2 = x^2 − (−5/3)x + 2/3 = x^2 + (5/3)x + 2/3 = 0.Multiply through by 3 to clear denominators: 3x^2 + 5x + 2 = 0.

Verification / Alternative check:
We can quickly check that if r is a root of 2x^2 + 5x + 3 = 0, then x = 1/r should satisfy 3x^2 + 5x + 2 = 0.Substitute x = 1/r into 3x^2 + 5x + 2 and multiply by r^2 to get 3 + 5r + 2r^2, which is just the original polynomial 2r^2 + 5r + 3 written in a different order. Since 2r^2 + 5r + 3 = 0, this confirms that 3x^2 + 5x + 2 = 0 has roots 1/r1 and 1/r2.

Why Other Options Are Wrong:
The coefficients in options B and C do not match the required sum and product of reciprocal roots. Their corresponding sums and products of roots differ from −5/3 and 2/3.Option D, "None", is incorrect because we have explicitly derived 3x^2 + 5x + 2 = 0 as the correct equation.

Common Pitfalls:
Some students incorrectly reverse the coefficients as c x^2 + b x + a without using Vieta relations carefully, which can happen to match only in some special cases.Another pitfall is forgetting to clear denominators after writing the new quadratic, leading to fractional coefficients that may not match any option even when the logic is correct.

Final Answer:
The required quadratic equation whose roots are the reciprocals of the roots of 2x^2 + 5x + 3 = 0 is 3x^2 + 5x + 2 = 0, corresponding to option A.