The equation `a "cos" x - "cos" 2x = 2a-7` passesses a solution if

Options

A. `a gt 6`
B. 2 le a le 6`
C. `a gt 2`
D. `a = -4`

Explanation

To solve the equation \( a \cos x - \cos 2x = 2a - 7 \) and determine the conditions under which it possesses a solution, we can follow these steps:

### Step 1: Rewrite the equation using the double angle formula
We know that \( \cos 2x = 2 \cos^2 x - 1 \). Therefore, we can rewrite the equation as:
\[ a \cos x - (2 \cos^2 x - 1) = 2a - 7 \]
This simplifies to:
\[ a \cos x - 2 \cos^2 x + 1 = 2a - 7 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ -2 \cos^2 x + a \cos x + 1 - 2a + 7 = 0 \]
This can be simplified to:
\[ -2 \cos^2 x + a \cos x + 8 - 2a = 0 \]
Multiplying through by -1, we get:
\[ 2 \cos^2 x - a \cos x + (2a - 8) = 0 \]

### Step 3: Identify the quadratic form
This is a quadratic equation in terms of \( \cos x \):
\[ 2 \cos^2 x - a \cos x + (2a - 8) = 0 \]

### Step 4: Use the quadratic formula
To find the roots of this quadratic equation, we can use the quadratic formula:
\[ \cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -a \), and \( c = 2a - 8 \). Plugging these values into the formula gives:
\[ \cos x = \frac{a \pm \sqrt{(-a)^2 - 4 \cdot 2 \cdot (2a - 8)}}{2 \cdot 2} \]
This simplifies to:
\[ \cos x = \frac{a \pm \sqrt{a^2 - 8a + 64}}{4} \]

### Step 5: Determine the conditions for solutions
For the equation to have real solutions for \( \cos x \), the discriminant must be non-negative:
\[ a^2 - 8a + 64 \geq 0 \]
Now, we can solve this inequality. The discriminant of this quadratic is:
\[ (-8)^2 - 4 \cdot 1 \cdot 64 = 64 - 256 = -192 \]
Since the discriminant is negative, the quadratic \( a^2 - 8a + 64 \) does not cross the x-axis and is always positive. Therefore, the expression is always non-negative.

### Conclusion
The equation \( a \cos x - \cos 2x = 2a - 7 \) possesses a solution for all values of \( a \).