Let x be the greater real root of 3x^2 + 8x + 4 = 0 and y be the greater real root of 4y^2 − 19y + 12 = 0. Compare x and y.

Question

Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context: Here we compare greater roots from two non-monic quadratics. Using the quadratic formula (or completing the square) gives exact values for comparison. Given Data / Assumptions: 3x^2 + 8x + 4 = 0. 4y^2 − 19y + 12 = 0. Concept / Approach: Compute each pair of roots. Identify the larger root in each case and compare numerically. Step-by-Step Solution: For x: Δ = 8^2 − 4*3*4 = 64 − 48 = 16; roots = (−8 ± 4)/(2*3) = {−2, −2/3}. Greater x = −2/3.For y: Δ = (−19)^2 − 4*4*12 = 361 − 192 = 169; roots = (19 ± 13)/(8) = {4, 3/4}. Greater y = 4.Thus x = −2/3 and y = 4 ⇒ x < y. Verification / Alternative check: Quick decimal comparison (−0.666… vs 4) confirms the inequality. Why Other Options Are Wrong: They contradict the numerical ordering. Common Pitfalls: Mixing up which of the two y-roots is larger or arithmetic slips in discriminant calculations. Final Answer: If x < y

Let x be the greater real root of 3x^2 + 8x + 4 = 0 and y be the greater real root of 4y^2 − 19y + 12 = 0. Compare x and y.

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