The nature of roots of the equation `2x^2 + sqrt(5x) - 1= 0` is ...........

Options

A. real and equal
B. imaginary and equal
C. imaginary and unequal
D. real and unequal

Answer

To determine the nature of the roots of the quadratic equation \(2x^2 + \sqrt{5}x - 1 = 0\), we can use the discriminant method. The discriminant (\(D\)) of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by:

\[ D = b^2 - 4ac \]

### Step 1: Identify the coefficients
From the equation \(2x^2 + \sqrt{5}x - 1 = 0\), we can identify the coefficients:
- \(a = 2\)
- \(b = \sqrt{5}\)
- \(c = -1\)

### Step 2: Calculate the discriminant
Now, we will substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:

\[ D = (\sqrt{5})^2 - 4 \cdot 2 \cdot (-1) \]

Calculating each term:

\[ D = 5 - 4 \cdot 2 \cdot (-1) \]
\[ D = 5 + 8 \]
\[ D = 13 \]

### Step 3: Determine the nature of the roots
The nature of the roots can be determined based on the value of the discriminant \(D\):
- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are complex (not real).

Since \(D = 13\) which is greater than 0, we conclude that the roots of the equation are real and distinct.

### Final Answer
The nature of the roots of the equation \(2x^2 + \sqrt{5}x - 1 = 0\) is **real and distinct**.

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