To determine which of the given options is a solution to the quadratic equation [tex]\(x^2 - 6x + 5 = 0\)[/tex], we can follow these steps:
1. Identify the quadratic equation: Our given equation is
[tex]\[
x^2 - 6x + 5 = 0
\][/tex]
2. Solve the quadratic equation: We will use the quadratic formula which is
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 5\)[/tex].
Plugging in these values, the quadratic formula becomes:
[tex]\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{6 \pm \sqrt{36 - 20}}{2}
\][/tex]
[tex]\[
x = \frac{6 \pm \sqrt{16}}{2}
\][/tex]
[tex]\[
x = \frac{6 \pm 4}{2}
\][/tex]
3. Calculate the roots: Now we can compute the values of [tex]\(x\)[/tex]:
[tex]\[
x = \frac{6 + 4}{2} = \frac{10}{2} = 5
\][/tex]
and
[tex]\[
x = \frac{6 - 4}{2} = \frac{2}{2} = 1
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 - 6x + 5 = 0\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = 1\)[/tex].
4. Verify which of the given options corresponds to these solutions:
- A. [tex]\(x = -5\)[/tex]
- B. [tex]\(x = -1\)[/tex]
- C. [tex]\(x = \frac{1}{5}\)[/tex]
- D. [tex]\(x = 5\)[/tex]
From the solutions [tex]\(x = 1\)[/tex] and [tex]\(x = 5\)[/tex], we see that:
- Option D is [tex]\(x = 5\)[/tex], which matches one of our solutions.
Therefore, the correct answer is:
[tex]\[
\boxed{5}
\][/tex]
This means the correct option among the given choices is:
[tex]\[
D. \ x = 5 \][/tex]
