To solve the quadratic equation [tex]\(x^2 - 16x + 64 = 0\)[/tex], we follow these steps:
### Step 1: Identify coefficients
The quadratic equation is in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -16\)[/tex]
- [tex]\(c = 64\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-16)^2 - 4 \cdot 1 \cdot 64
\][/tex]
[tex]\[
\Delta = 256 - 256
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
### Step 3: Determine the nature of the roots
Since the discriminant [tex]\(\Delta\)[/tex] is 0, this indicates that the quadratic equation has exactly one real root (a repeated root).
### Step 4: Calculate the root(s)
The roots of the quadratic equation are given by the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\(\Delta = 0\)[/tex], the formula simplifies to:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x = \frac{-(-16)}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{16}{2}
\][/tex]
[tex]\[
x = 8
\][/tex]
Therefore, the quadratic equation [tex]\(x^2 - 16x + 64 = 0\)[/tex] has a double root at [tex]\(x = 8\)[/tex].
### Step 5: Choose the correct answer
From the given options:
- A. [tex]\(x = 4\)[/tex] and [tex]\(x = 16\)[/tex]
- B. [tex]\(x = 8\)[/tex] only
- C. [tex]\(x = 8\)[/tex] and [tex]\(x = -8\)[/tex]
- D. [tex]\(x = -2\)[/tex] and [tex]\(x = 32\)[/tex]
The correct answer is B. [tex]\(x = 8\)[/tex] only.
