For which of the following quadratic equations do the roots remain real numbers?

Introduction / Context:
This question checks understanding of when a quadratic equation has real roots. The condition for real roots is based on the discriminant, a fundamental concept in algebra and quadratic equations used widely in aptitude tests and school examinations.

Given Data / Assumptions:

• General quadratic equation: ax^2 + bx + c = 0.
• Options are different quadratic equations.
• We must identify which one has real roots.

Concept / Approach:
For ax^2 + bx + c = 0, the discriminant D is given by D = b^2 - 4ac.

• If D > 0, there are two distinct real roots.
• If D = 0, there is one repeated real root.
• If D < 0, the roots are complex and not real.

We simply compute D for each option and check its sign.

Step-by-Step Solution:
Option A: 4x^2 - 9x + 6 = 0, here a = 4, b = -9, c = 6. D = (-9)^2 - 4*4*6 = 81 - 96 = -15 (negative, no real roots). Option B: 3x^2 - 2x + 6 = 0, a = 3, b = -2, c = 6. D = (-2)^2 - 4*3*6 = 4 - 72 = -68 (negative, no real roots). Option C: 2x^2 - 7x + 6 = 0, a = 2, b = -7, c = 6. D = (-7)^2 - 4*2*6 = 49 - 48 = 1 (positive, two distinct real roots). Option D: x^2 - 2x + 2 = 0, a = 1, b = -2, c = 2. D = (-2)^2 - 4*1*2 = 4 - 8 = -4 (negative, no real roots). Option E: x^2 + 4x + 5 = 0, a = 1, b = 4, c = 5. D = 4^2 - 4*1*5 = 16 - 20 = -4 (negative, no real roots). Hence only 2x^2 - 7x + 6 = 0 has real roots.

Verification / Alternative check:
We can factor option C: 2x^2 - 7x + 6 = 0 as (2x - 3)(x - 2) = 0. This gives real roots x = 3/2 and x = 2, which confirms that the discriminant result is correct. None of the other equations factor nicely into real linear factors.

Why Other Options Are Wrong:
Options A, B, D and E all have negative discriminants, which means their roots are complex conjugates, not real numbers. Even though coefficients look simple, a negative discriminant disqualifies them immediately for real roots.

Common Pitfalls:
Many learners guess based on factorisation attempts without computing the discriminant. This can be risky when factorisation is not obvious. Always remember that the discriminant test is decisive and quick, especially in multiple choice exams where you must justify real versus complex roots efficiently.

Final Answer:
The quadratic equation that has real roots is 2x^2 - 7x + 6 = 0.