If all real values of x obtained from the equation `4^(x) - (a-3)2^(x) + a - 4 = 0` are non-positive, then a lies in

Options

A. (4, 5]
B. (0, 4)
C. `(4, oo)`
D. none of these

Explanation

To solve the equation \( 4^x - (a-3)2^x + a - 4 = 0 \) and determine the range of \( a \) such that all real values of \( x \) are non-positive, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 4^x - (a-3)2^x + a - 4 = 0 \]
We know that \( 4^x = (2^x)^2 \), so we can substitute \( y = 2^x \). This gives us:
\[ y^2 - (a-3)y + (a - 4) = 0 \]

### Step 2: Analyze the quadratic equation
The equation is now a quadratic in terms of \( y \):
\[ y^2 - (a-3)y + (a - 4) = 0 \]
For the quadratic equation \( Ay^2 + By + C = 0 \) to have real roots, the discriminant must be non-negative:
\[ D = B^2 - 4AC \]
Here, \( A = 1 \), \( B = -(a-3) \), and \( C = a - 4 \). Therefore, the discriminant \( D \) is:
\[ D = (-(a-3))^2 - 4 \cdot 1 \cdot (a - 4) \]
\[ D = (a-3)^2 - 4(a - 4) \]

### Step 3: Simplify the discriminant
Now, we simplify the discriminant:
\[ D = (a-3)^2 - 4a + 16 \]
Expanding \( (a-3)^2 \):
\[ D = a^2 - 6a + 9 - 4a + 16 \]
\[ D = a^2 - 10a + 25 \]
We can factor this as:
\[ D = (a-5)^2 \]

### Step 4: Condition for non-positive roots
For the roots to be real and non-positive, we need \( D \geq 0 \) (which is always true since it's a square) and the roots must be less than or equal to zero. The roots of the quadratic equation can be found using the quadratic formula:
\[ y = \frac{(a-3) \pm \sqrt{D}}{2} \]
Since \( D = (a-5)^2 \), the roots are:
\[ y = \frac{(a-3) \pm (a-5)}{2} \]
Calculating the two roots:
1. \( y_1 = \frac{(a-3) + (a-5)}{2} = \frac{2a - 8}{2} = a - 4 \)
2. \( y_2 = \frac{(a-3) - (a-5)}{2} = \frac{2}{2} = 1 \)

### Step 5: Set conditions for non-positive \( x \)
Since \( y = 2^x \) and we want \( x \leq 0 \), we require:
\[ y \leq 1 \]
This gives us two conditions:
1. \( a - 4 \leq 1 \) (for the first root)
2. \( 1 \leq 1 \) (for the second root, which is always true)

From \( a - 4 \leq 1 \):
\[ a \leq 5 \]

### Step 6: Determine the lower bound for \( a \)
Since \( y = 2^x \) must be positive, we also require:
\[ a - 4 > 0 \implies a > 4 \]

### Conclusion
Combining both conditions, we find:
\[ 4 < a \leq 5 \]
Thus, the range of \( a \) is:
\[ (4, 5] \]

### Final Answer
The correct option is **(4, 5]**.