The number of roots of the quadratic equation 8 `sec^(2) theta-6 sec theta+1=0` is

Options

A. infinite
B. 1
C. 2
D. 0

Answer

To determine the number of roots of the quadratic equation \( 8 \sec^2 \theta - 6 \sec \theta + 1 = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \). Here, we let \( x = \sec \theta \):
- \( a = 8 \)
- \( b = -6 \)
- \( c = 1 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-6)^2 - 4 \cdot 8 \cdot 1 = 36 - 32 = 4 \]

### Step 3: Analyze the discriminant
The value of the discriminant helps us determine the nature of the roots:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is one real root (a repeated root).
- If \( D < 0 \), there are no real roots.

In our case, \( D = 4 > 0 \), which means there are two distinct real roots.

### Step 4: Find the roots
To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-6) \pm \sqrt{4}}{2 \cdot 8} = \frac{6 \pm 2}{16} \]
This gives us:
\[ x_1 = \frac{8}{16} = \frac{1}{2}, \quad x_2 = \frac{4}{16} = \frac{1}{4} \]

### Step 5: Check the validity of the roots
Since \( x = \sec \theta \), we need to check if \( \sec \theta = \frac{1}{2} \) and \( \sec \theta = \frac{1}{4} \) are valid:
- The secant function \( \sec \theta \) is defined as \( \sec \theta = \frac{1}{\cos \theta} \).
- The values \( \sec \theta = \frac{1}{2} \) and \( \sec \theta = \frac{1}{4} \) imply that \( \cos \theta = 2 \) and \( \cos \theta = 4 \), respectively. Both of these values are not possible since the range of \( \cos \theta \) is \([-1, 1]\).

### Conclusion
Since neither of the roots \( \frac{1}{2} \) nor \( \frac{1}{4} \) are valid for \( \sec \theta \), we conclude that there are no real roots for the given quadratic equation.

Thus, the number of roots of the quadratic equation \( 8 \sec^2 \theta - 6 \sec \theta + 1 = 0 \) is:
\[ \text{Number of roots} = 0 \]

### Final Answer
The number of roots is \( \boxed{0} \).

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