The quadratic equation `2x^(2) - sqrt5 x +1 = 0` has

Options

A. two distinct real roots
B. two equal real roots
C. no real roots
D. more than 2 real roots

Answer

To determine the nature of the roots of the quadratic equation \(2x^2 - \sqrt{5}x + 1 = 0\), we will follow these steps:

### Step 1: Identify the coefficients
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). From the given equation, we can identify:
- \(a = 2\)
- \(b = -\sqrt{5}\)
- \(c = 1\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-\sqrt{5})^2 - 4 \cdot 2 \cdot 1 \]
Calculating \(b^2\):
\[ D = 5 - 4 \cdot 2 \cdot 1 \]
Calculating \(4ac\):
\[ D = 5 - 8 \]
Thus, we find:
\[ D = -3 \]

### Step 3: Analyze the discriminant
Now, we analyze the value of the discriminant:
- If \(D > 0\), the equation has two distinct real roots.
- If \(D = 0\), the equation has two equal real roots.
- If \(D < 0\), the equation has no real roots (the roots are imaginary).

Since \(D = -3\), which is less than zero, we conclude that:

### Conclusion
The quadratic equation \(2x^2 - \sqrt{5}x + 1 = 0\) has no real roots.

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