To determine which of the given options is a solution to the equation [tex]\(x^2 - 6x + 5 = 0\)[/tex], we should consider the general process for solving a quadratic equation. Here’s a step-by-step solution:
1. Identify the quadratic equation:
The equation provided is [tex]\(x^2 - 6x + 5 = 0\)[/tex].
2. Use the quadratic formula:
The quadratic formula allows us to find the roots (solutions) of any quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex].
The formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\(x^2 - 6x + 5 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -6\)[/tex]
- [tex]\(c = 5\)[/tex]
3. Calculate the discriminant:
The discriminant [tex]\(D\)[/tex] of a quadratic equation is:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substituting the values:
[tex]\[
D = (-6)^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16
\][/tex]
4. Find the roots using the quadratic formula:
Substituting the values into the quadratic formula:
[tex]\[
x = \frac{-(-6) \pm \sqrt{16}}{2 \cdot 1}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{6 \pm 4}{2}
\][/tex]
5. Calculate the two potential roots:
[tex]\[
x = \frac{6 + 4}{2} = \frac{10}{2} = 5
\][/tex]
[tex]\[
x = \frac{6 - 4}{2} = \frac{2}{2} = 1
\][/tex]
6. Identify the solutions:
The solutions to the quadratic equation [tex]\(x^2 - 6x + 5 = 0\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = 1\)[/tex].
7. Match the solutions with the given options:
We now check which of the given options matches our solutions:
- [tex]\(x = -5\)[/tex]
- [tex]\(x = -1\)[/tex]
- [tex]\(x = \frac{1}{5}\)[/tex]
- [tex]\(x = 5\)[/tex]
The solution [tex]\(x = 5\)[/tex] matches option D.
Therefore, the correct answer is:
[tex]\[
\boxed{5}
\][/tex]
