Among the following quadratic equations, which one has real roots, that is, a non negative discriminant?

Introduction / Context:
This aptitude question tests the concept of the discriminant of a quadratic equation and how it determines the nature of roots. For an equation ax^2 + bx + c = 0, the discriminant b^2 - 4ac indicates whether the roots are real, equal, or complex. Many competitive exams ask you to quickly inspect several equations and identify which one has real solutions.

Given Data / Assumptions:

• We are given four quadratic equations in x.
• We must find which equation has real roots.
• Real roots occur when the discriminant b^2 - 4ac is greater than or equal to zero.

Concept / Approach:
For a quadratic equation ax^2 + bx + c = 0:

• If b^2 - 4ac > 0, roots are real and distinct.
• If b^2 - 4ac = 0, roots are real and equal.
• If b^2 - 4ac < 0, roots are complex and non real.

We compute the discriminant of each option and select the equation where the discriminant is non negative.

Step-by-Step Solution:
Option a: 4x^2 - 3x + 6 = 0, here a = 4, b = -3, c = 6 Discriminant D = b^2 - 4ac = (-3)^2 - 4*4*6 = 9 - 96 = -87 (negative, no real roots) Option b: 2x^2 + 7x + 6 = 0, here a = 2, b = 7, c = 6 D = 7^2 - 4*2*6 = 49 - 48 = 1 (positive, two real distinct roots) Option c: x^2 - 2x + 4 = 0, a = 1, b = -2, c = 4 D = (-2)^2 - 4*1*4 = 4 - 16 = -12 (negative, complex roots) Option d: 3x^2 - 4x + 3 = 0, a = 3, b = -4, c = 3 D = (-4)^2 - 4*3*3 = 16 - 36 = -20 (negative, complex roots) Only option b has a positive discriminant, so it has real roots.

Verification / Alternative check:
We can quickly factor option b to double check. Try to write 2x^2 + 7x + 6 as (2x + 3)(x + 2). Multiply to get 2x^2 + 4x + 3x + 6 = 2x^2 + 7x + 6, which matches the original. Since it factors into linear factors with real coefficients, it clearly has two real roots, confirming the discriminant approach.

Why Other Options Are Wrong:
In options a, c, and d the discriminant is negative, which means the roots are complex conjugates, not real numbers. Even if you tried to factor them, you would not find real linear factors, which is consistent with the discriminant test.

Common Pitfalls:
Students sometimes confuse the sign pattern in b^2 - 4ac and mistakenly compute 4ac with the wrong sign. Another common error is to forget that any negative discriminant, even a small one, means there are no real roots. Some candidates also try to factor first, which can waste time when the discriminant test is much faster.

Final Answer:
The only quadratic equation with real roots is 2x^2 + 7x + 6 = 0.