The roots of the equation `x^2 – 3x – 9 = 0` are :

Options

A. real and unequal
B. real and equal
C. roots are not equal
D. imaginary roots

Answer

To find the roots of the quadratic equation \( x^2 - 3x - 9 = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we identify:
- \( a = 1 \)
- \( b = -3 \)
- \( c = -9 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-3)^2 - 4 \cdot 1 \cdot (-9) \]
\[ D = 9 + 36 \]
\[ D = 45 \]

### Step 3: Analyze the discriminant
Since \( D > 0 \), this indicates that the roots of the equation are real and unequal.

### Step 4: Find the roots using the quadratic formula
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 \( a \):
\[ x = \frac{-(-3) \pm \sqrt{45}}{2 \cdot 1} \]
\[ x = \frac{3 \pm \sqrt{45}}{2} \]

### Step 5: Simplify the square root
We can simplify \( \sqrt{45} \):
\[ \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \]
Thus, substituting back:
\[ x = \frac{3 \pm 3\sqrt{5}}{2} \]

### Step 6: Write the final roots
The roots of the equation are:
\[ x_1 = \frac{3 + 3\sqrt{5}}{2}, \quad x_2 = \frac{3 - 3\sqrt{5}}{2} \]

### Summary of the roots
The roots of the equation \( x^2 - 3x - 9 = 0 \) are:
\[ x_1 = \frac{3 + 3\sqrt{5}}{2}, \quad x_2 = \frac{3 - 3\sqrt{5}}{2} \]