The equation `ax^(4)-2x^(2)-(a-1)=0` will have real and unequal roots if

Options

A. `o lt a lt 1`
B. `a gt 0`, `a ne 1`
C. `a lt 0` , `a ne 1`
D. none of these

Explanation

To determine the conditions under which the equation \( ax^4 - 2x^2 - (a - 1) = 0 \) has real and unequal roots, we can follow these steps:

### Step 1: Substitute \( x^2 \) with \( y \)
Let \( y = x^2 \). Then, the equation becomes:
\[ ay^2 - 2y - (a - 1) = 0 \]
This is a quadratic equation in terms of \( y \).

### Step 2: Identify the discriminant
For the quadratic equation \( ay^2 - 2y - (a - 1) = 0 \) to have real and unequal roots, the discriminant must be greater than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = a \), \( b = -2 \), and \( c = -(a - 1) \). Thus,
\[ D = (-2)^2 - 4(a)(-(a - 1)) = 4 + 4a(a - 1) \]
Simplifying this gives:
\[ D = 4 + 4a^2 - 4a = 4a^2 - 4a + 4 \]

### Step 3: Set the discriminant greater than zero
We need:
\[ 4a^2 - 4a + 4 > 0 \]
This can be simplified to:
\[ a^2 - a + 1 > 0 \]

### Step 4: Analyze the quadratic inequality
The quadratic \( a^2 - a + 1 \) has a discriminant:
\[ D = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]
Since the discriminant is negative, the quadratic \( a^2 - a + 1 \) does not have any real roots and is always positive for all real values of \( a \).

### Step 5: Ensure the roots are positive
Next, we need to ensure that the roots \( y \) (which correspond to \( x^2 \)) are positive. The sum and product of the roots of the quadratic equation \( ay^2 - 2y - (a - 1) = 0 \) can be given by Vieta's formulas:
- Sum of roots \( y_1 + y_2 = \frac{2}{a} \)
- Product of roots \( y_1y_2 = \frac{-(a - 1)}{a} \)

For the roots to be positive:
1. The sum \( \frac{2}{a} > 0 \) implies \( a > 0 \).
2. The product \( \frac{-(a - 1)}{a} > 0 \) implies \( a - 1 < 0 \) or \( a < 1 \).

### Step 6: Combine the conditions
Combining the conditions \( a > 0 \) and \( a < 1 \), we find:
\[ 0 < a < 1 \]

### Conclusion
Thus, the equation \( ax^4 - 2x^2 - (a - 1) = 0 \) will have real and unequal roots if \( a \) is in the range \( (0, 1) \).