Question

What is the nature of the roots of the quadratic equation \(2x^2 - 6x + 3 = 0\) ?

• A. real and unequal
• B. real and equal
• C. not real
• D. none of these

Hint:

The discriminant is a powerful tool. You don't need to solve the equation to know the type of roots it has. Just calculating \(b^2 - 4ac\) is enough.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is determined by its discriminant, \(D = b^2 - 4ac\).
- If \(D > 0\), the roots are real and unequal.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are not real (they are complex).

Step 2: Key Formula or Approach:
Calculate the discriminant \(D = b^2 - 4ac\) and analyze its value.

Step 3: Detailed Explanation:
For the equation \(2x^2 - 6x + 3 = 0\), we have:
\(a = 2\), \(b = -6\), \(c = 3\).
Now, calculate the discriminant:
\[ D = (-6)^2 - 4(2)(3) \] \[ D = 36 - 24 \] \[ D = 12 \] Since \(D = 12\), which is greater than 0, the roots of the equation are real and unequal.

Step 4: Final Answer:
The nature of the roots is real and unequal.