The roots of the equation `x^(2) - 8x + 16 = 0`.

Options

A. Are imaginary
B. Are distinct and real
C. Are equal and real
D. Cannot be determined

Answer

To find the roots of the quadratic equation \(x^2 - 8x + 16 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 1\)
- \(b = -8\)
- \(c = 16\)

### Step 2: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-8)^2 - 4 \cdot 1 \cdot 16 \]
\[ D = 64 - 64 = 0 \]

### Step 3: Determine the nature of the roots
Since the discriminant \(D = 0\), this indicates that the quadratic equation has real and equal roots.

### Step 4: Calculate the roots using the quadratic formula
The roots of the quadratic equation can be found using the formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-8) \pm \sqrt{0}}{2 \cdot 1} \]
\[ x = \frac{8 \pm 0}{2} \]
\[ x = \frac{8}{2} = 4 \]

### Conclusion
The roots of the equation \(x^2 - 8x + 16 = 0\) are:
\[ x = 4 \quad \text{(with multiplicity 2)} \]

### Summary of the nature of the roots
- The roots are real and equal.

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