If `c lt a lt b lt d`, then roots of the equation `bx^(2)+(1-b(c+d)x+bcd-a=0`

Options

A. are real and one lies between `c` and `a`
B. are real and distinct in which one lies between `a` and `b`
C. are real and distinct in which one lies between `c` and `d`
D. are not real

Explanation

To solve the equation \( bx^2 + (1 - b(c + d))x + (bcd - a) = 0 \) under the condition \( c < a < b < d \), we can follow these steps:

### Step 1: Write the equation in standard form
The given equation is already in standard quadratic form:
\[ bx^2 + (1 - b(c + d))x + (bcd - a) = 0 \]

### Step 2: Identify coefficients
From the standard form \( Ax^2 + Bx + C = 0 \), we identify:
- \( A = b \)
- \( B = 1 - b(c + d) \)
- \( C = bcd - a \)

### Step 3: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (1 - b(c + d))^2 - 4b(bcd - a) \]

### Step 4: Simplify the discriminant
Expanding \( D \):
\[ D = (1 - b(c + d))^2 - 4b^2cd + 4ab \]
\[ = 1 - 2b(c + d) + b^2(c + d)^2 - 4b^2cd + 4ab \]

### Step 5: Analyze the discriminant
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ D > 0 \]

### Step 6: Determine the nature of the roots
Given the conditions \( c < a < b < d \):
- Since \( c < a \), we have \( c - a < 0 \).
- Since \( a < d \), we have \( d - a > 0 \).

This indicates that one root will be negative and the other will be positive, confirming that the roots are real and distinct.

### Step 7: Identify the intervals of the roots
Using Vieta's formulas, the sum and product of the roots can be analyzed:
- The roots can be expressed as \( r_1 \) and \( r_2 \).
- Since one root is negative and the other is positive, we can conclude that one root lies between \( c \) and \( d \).

### Conclusion
Thus, the roots of the equation are real and distinct, with one root lying between \( c \) and \( d \).