Solve: `sqrt(5)x^(2) + x + sqrt(5) = 0`

Explanation

To solve the quadratic equation \( \sqrt{5}x^2 + x + \sqrt{5} = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( Ax^2 + Bx + C = 0 \). Here, we can identify:
- \( A = \sqrt{5} \)
- \( B = 1 \)
- \( C = \sqrt{5} \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = 1^2 - 4(\sqrt{5})(\sqrt{5}) = 1 - 4 \cdot 5 = 1 - 20 = -19 \]

### Step 3: Analyze the discriminant
Since \( D < 0 \), this indicates that the roots of the quadratic equation are complex.

### Step 4: Use the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{D}}{2A} \]
Substituting the values of \( B \), \( D \), and \( A \):
\[ x = \frac{-1 \pm \sqrt{-19}}{2\sqrt{5}} \]

### Step 5: Simplify the square root of the discriminant
The square root of a negative number can be expressed using \( i \) (where \( i = \sqrt{-1} \)):
\[ \sqrt{-19} = \sqrt{19}i \]
Thus, we can rewrite the expression for \( x \):
\[ x = \frac{-1 \pm \sqrt{19}i}{2\sqrt{5}} \]

### Step 6: Finalize the roots
The final expression for the roots is:
\[ x = \frac{-1}{2\sqrt{5}} \pm \frac{\sqrt{19}}{2\sqrt{5}}i \]

### Summary of the solution
The roots of the equation \( \sqrt{5}x^2 + x + \sqrt{5} = 0 \) are:
\[ x = \frac{-1}{2\sqrt{5}} + \frac{\sqrt{19}}{2\sqrt{5}}i \quad \text{and} \quad x = \frac{-1}{2\sqrt{5}} - \frac{\sqrt{19}}{2\sqrt{5}}i \]

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