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. canot be ascertained

Explanation

To find the roots of the 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 \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -8 \)
- \( c = 16 \)

### 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 \( c \):
\[ D = (-8)^2 - 4 \cdot 1 \cdot 16 \]
Calculating \( D \):
\[ D = 64 - 64 = 0 \]

### Step 3: Determine the nature of the roots
The nature of the roots can be determined based on the value of the discriminant:
- If \( D > 0 \), the roots are real and distinct.
- If \( D = 0 \), the roots are real and equal.
- If \( D < 0 \), the roots are imaginary.

Since we found \( D = 0 \), the roots of the equation are real and equal.

### Step 4: Find the roots using 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:
\[ x = \frac{-(-8) \pm \sqrt{0}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{8 \pm 0}{2} = \frac{8}{2} = 4 \]
Thus, the roots are \( x = 4 \) (with multiplicity 2).

### Conclusion
The roots of the equation \( x^2 - 8x + 16 = 0 \) are real and equal, specifically \( x = 4 \).

---