The quadratic equation whose one root is ` - (i)/(4)` :

Options

A. (a) ` x^(2) -(1)/(4)= 0`
B. (b) ` x^(2) +(1)/(4) =0`
C. (c) ` x^(2) +(1)/(16) =0`
D. (d) ` x^(2) +(1)/(8) =0 `

Answer

To find the quadratic equation whose one root is \(-\frac{i}{4}\), we can follow these steps:

### Step 1: Identify the roots
Since one root is given as \(-\frac{i}{4}\), we denote this root as \(a = -\frac{i}{4}\). The other root, being the conjugate of the first root (as complex roots come in conjugate pairs), will be \(b = \frac{i}{4}\).

### Step 2: Write the quadratic equation
The quadratic equation can be expressed in terms of its roots \(a\) and \(b\) using the formula:
\[ (x - a)(x - b) = 0 \]
Substituting the values of \(a\) and \(b\):
\[ (x - (-\frac{i}{4}))(x - \frac{i}{4}) = 0 \]
This simplifies to:
\[ (x + \frac{i}{4})(x - \frac{i}{4}) = 0 \]

### Step 3: Apply the difference of squares formula
The expression \((x + \frac{i}{4})(x - \frac{i}{4})\) can be simplified using the difference of squares formula:
\[ a^2 - b^2 \]
where \(a = x\) and \(b = \frac{i}{4}\). Thus, we have:
\[ x^2 - \left(\frac{i}{4}\right)^2 = 0 \]

### Step 4: Calculate \(\left(\frac{i}{4}\right)^2\)
Calculating \(\left(\frac{i}{4}\right)^2\):
\[ \left(\frac{i}{4}\right)^2 = \frac{i^2}{16} = \frac{-1}{16} \]
since \(i^2 = -1\).

### Step 5: Substitute back into the equation
Substituting this back into the equation gives:
\[ x^2 - \left(-\frac{1}{16}\right) = 0 \]
which simplifies to:
\[ x^2 + \frac{1}{16} = 0 \]

### Conclusion
Thus, the quadratic equation whose one root is \(-\frac{i}{4}\) is:
\[ x^2 + \frac{1}{16} = 0 \]

### Final Answer
The correct option is:
**Option C: \(x^2 + \frac{1}{16} = 0\)**