To determine which of the given options is a solution to the quadratic equation [tex]\(x^2 - 6x + 5 = 0\)[/tex], we'll follow these steps:
### 1. Identify the coefficients in the quadratic equation.
The given equation [tex]\(x^2 - 6x + 5 = 0\)[/tex] has the following coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = 5 \)[/tex]
### 2. Calculate the discriminant.
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is calculated using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 5\)[/tex] into the formula:
[tex]\[ \Delta = (-6)^2 - 4 \cdot 1 \cdot 5 \][/tex]
[tex]\[ \Delta = 36 - 20 \][/tex]
[tex]\[ \Delta = 16 \][/tex]
### 3. Determine if the discriminant is positive, negative, or zero.
Since the discriminant [tex]\(\Delta = 16\)[/tex] is positive, there are two distinct real solutions to the quadratic equation.
### 4. Use the quadratic formula to find the solutions.
The solutions of the quadratic equation are given by the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(\Delta = 16\)[/tex], [tex]\(a = 1\)[/tex], and [tex]\(b = -6\)[/tex] into the formula:
[tex]\[ x = \frac{-(-6) \pm \sqrt{16}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{6 \pm 4}{2} \][/tex]
There are two solutions:
- [tex]\( x_1 = \frac{6 + 4}{2} = \frac{10}{2} = 5 \)[/tex]
- [tex]\( x_2 = \frac{6 - 4}{2} = \frac{2}{2} = 1 \)[/tex]
### 5. Analyze the given options.
We need to determine which of the provided options is a correct solution to the equation.
The provided options are:
A. [tex]\( x = -5 \)[/tex]
B. [tex]\( x = -1 \)[/tex]
C. [tex]\( x = \frac{1}{5} \)[/tex]
D. [tex]\( x = 5 \)[/tex]
From our solutions [tex]\(x_1 = 5\)[/tex] and [tex]\(x_2 = 1\)[/tex], we see that the correct answer among the options is:
[tex]\[ \boxed{x = 5} \][/tex]
Hence, option D is the solution.
