What is the difference in the roots of the equation `x^(2) - 10x + 9 = 0`.

Options

A. 2
B. 3
C. 5
D. 8

Answer

To find the difference in the roots of the quadratic equation \(x^2 - 10x + 9 = 0\), 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,
- \(a = 1\)
- \(b = -10\)
- \(c = 9\)

### 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 \(c\):
\[ D = (-10)^2 - 4 \cdot 1 \cdot 9 \]
\[ D = 100 - 36 \]
\[ D = 64 \]

### Step 3: Find the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \(b\), \(D\), and \(a\):
\[ x = \frac{-(-10) \pm \sqrt{64}}{2 \cdot 1} \]
\[ x = \frac{10 \pm 8}{2} \]

### Step 4: Calculate the two roots
Now, we can calculate the two roots:
1. First root:
\[ x_1 = \frac{10 + 8}{2} = \frac{18}{2} = 9 \]
2. Second root:
\[ x_2 = \frac{10 - 8}{2} = \frac{2}{2} = 1 \]

### Step 5: Calculate the difference in the roots
The difference in the roots is given by:
\[ \text{Difference} = x_1 - x_2 = 9 - 1 = 8 \]

### Final Answer
The difference in the roots of the equation \(x^2 - 10x + 9 = 0\) is \(8\).

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