Quadratic equation `8 "sec"^(2) theta - 6 "sec" theta + 1 = 0` has

Options

A. exactlyk two roots
B. exactly four roots
C. infinitely many roots
D. no roots

Answer

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

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 8 \)
- \( b = -6 \)
- \( c = 1 \)

### Step 2: Use the quadratic formula
The quadratic formula to find the roots of the equation is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, we will substitute \( \sec \theta \) for \( x \).

### Step 3: Substitute the coefficients into the formula
Substituting the values of \( a \), \( b \), and \( c \):
\[ \sec \theta = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 8 \cdot 1}}{2 \cdot 8} \]
This simplifies to:
\[ \sec \theta = \frac{6 \pm \sqrt{36 - 32}}{16} \]

### Step 4: Simplify the expression under the square root
Calculating the discriminant:
\[ \sec \theta = \frac{6 \pm \sqrt{4}}{16} \]
This simplifies to:
\[ \sec \theta = \frac{6 \pm 2}{16} \]

### Step 5: Calculate the two possible values for \( \sec \theta \)
Now we can find the two possible values:
1. \( \sec \theta = \frac{6 + 2}{16} = \frac{8}{16} = \frac{1}{2} \)
2. \( \sec \theta = \frac{6 - 2}{16} = \frac{4}{16} = \frac{1}{4} \)

### Step 6: Convert secant to cosine
Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we can write:
1. \( \frac{1}{\cos \theta} = \frac{1}{2} \) implies \( \cos \theta = 2 \)
2. \( \frac{1}{\cos \theta} = \frac{1}{4} \) implies \( \cos \theta = 4 \)

### Step 7: Analyze the solutions
However, we know that the range of \( \cos \theta \) is between -1 and 1. Therefore:
- \( \cos \theta = 2 \) is not possible.
- \( \cos \theta = 4 \) is also not possible.

### Conclusion
Since both values of \( \sec \theta \) lead to impossible values for \( \cos \theta \), we conclude that there are no solutions to the equation. Thus, the quadratic equation \( 8 \sec^2 \theta - 6 \sec \theta + 1 = 0 \) has no roots.

### Final Answer
**No solution exists.**

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