In quadratic equations, determine which of the following equations has real and distinct roots, that is, two different real solutions for x.

Introduction / Context:
This question checks your understanding of the discriminant of a quadratic equation and how it determines the nature of the roots. Quadratic equations can have two distinct real roots, one repeated real root, or a pair of complex conjugate roots. Knowing how to quickly classify them using the discriminant is a standard skill in algebra and aptitude exams.

Given Data / Assumptions:

• Each option is a quadratic equation of the form ax^2 + bx + c = 0.
• We must find which equation has two distinct real roots.
• The discriminant is defined as D = b^2 − 4ac.
• Real and distinct roots occur when D > 0.

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

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has real and equal (repeated) roots.
• If D < 0, the equation has no real roots (complex roots).

We will compute the discriminant for each equation and check which one yields a positive value. This approach allows us to decide without actually solving the equations for x.

Step-by-Step Solution:
1) For option a: 3x^2 − 6x + 2 = 0, we have a = 3, b = −6, c = 2. 2) Compute the discriminant: D = (−6)^2 − 4 * 3 * 2 = 36 − 24 = 12. This is positive, so roots are real and distinct. 3) For option b: 3x^2 − 6x + 3 = 0, a = 3, b = −6, c = 3. Then D = (−6)^2 − 4 * 3 * 3 = 36 − 36 = 0, so roots are real and equal. 4) For option c: x^2 − 8x + 16 = 0, a = 1, b = −8, c = 16. D = (−8)^2 − 4 * 1 * 16 = 64 − 64 = 0, real and equal roots. 5) For option d: 4x^2 − 8x + 4 = 0, a = 4, b = −8, c = 4. D = (−8)^2 − 4 * 4 * 4 = 64 − 64 = 0, real and equal roots. 6) Only option a has D > 0, so only that equation has two distinct real roots.

Verification / Alternative check:
We can also factor or use the quadratic formula for option a. Using the quadratic formula for 3x^2 − 6x + 2 = 0 gives x = [6 ± √12] / 6 = [6 ± 2√3] / 6 = 1 ± (√3)/3, which clearly yields two different real values. For options b, c, and d, the repeated roots can be seen directly: for example, x^2 − 8x + 16 = (x − 4)^2 and 4x^2 − 8x + 4 = 4(x − 1)^2, each having a single repeated root.

Why Other Options Are Wrong:
Options b, c, and d all have discriminant equal to zero, meaning they have real but equal roots, not distinct. Option e, 'None of these', is incorrect because we have already identified that option a does indeed have two distinct real roots. Therefore, only option a satisfies the required condition of real and distinct roots.

Common Pitfalls:
Students often miscompute the discriminant by forgetting the factor 4ac or by mishandling the sign of b. Another common mistake is assuming that the presence of a square in the constant term automatically implies equal roots, which is not always correct. Always compute D carefully, and remember that only D > 0 guarantees two different real solutions.

Final Answer:
The quadratic equation with real and distinct roots is 3x^2 - 6x + 2 = 0.