Solve the following quadratic: `4x^2+1=0`

Explanation

To solve the quadratic equation \(4x^2 + 1 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 4x^2 + 1 = 0 \]

### Step 2: Isolate the quadratic term
Subtract 1 from both sides:
\[ 4x^2 = -1 \]

### Step 3: Divide by 4
Divide both sides by 4 to isolate \(x^2\):
\[ x^2 = -\frac{1}{4} \]

### Step 4: Take the square root
To solve for \(x\), take the square root of both sides. Remember that the square root of a negative number involves the imaginary unit \(i\):
\[ x = \pm \sqrt{-\frac{1}{4}} = \pm \frac{\sqrt{-1}}{\sqrt{4}} = \pm \frac{i}{2} \]

### Step 5: Write the final solutions
Thus, the solutions to the equation are:
\[ x = \frac{i}{2} \quad \text{and} \quad x = -\frac{i}{2} \]

### Summary of the solutions
The roots of the quadratic equation \(4x^2 + 1 = 0\) are:
\[ x = \frac{i}{2}, \quad x = -\frac{i}{2} \]

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