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.

Difficulty: Medium

If x > y
If x ≥ y
If x < y
If x ≤ y
If x = y

Correct Answer: If x < y

Explanation:

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.

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.

More Questions from Quadratic Equation

Discussion & Comments