If `a x^2+(b-c)x+a-b-c=0` has unequal real roots for all `c in R ,t h e n` `b<0 a >0`

Options

A. `blt0lta`
B. `altoltb`
C. `bltalt0`
D. `bgtagt0`

Explanation

To solve the problem, we need to analyze the quadratic equation given by:

\[ ax^2 + (b - c)x + (a - b - c) = 0 \]

We want to determine the conditions under which this equation has unequal real roots for all \( c \in \mathbb{R} \).

### Step 1: Identify the condition for unequal real roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \) to have unequal real roots, the discriminant must be positive:

\[ D = B^2 - 4AC > 0 \]

In our case:
- \( A = a \)
- \( B = b - c \)
- \( C = a - b - c \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by:

\[ D = (b - c)^2 - 4a(a - b - c) \]

Expanding this, we have:

\[ D = (b^2 - 2bc + c^2) - 4a(a - b - c) \]

\[ D = b^2 - 2bc + c^2 - 4a^2 + 4ab + 4ac \]

### Step 3: Rearranging the discriminant
Rearranging gives us:

\[ D = c^2 + (4a - 2b)c + (b^2 - 4a^2 + 4ab) \]

### Step 4: Condition for \( D > 0 \) for all \( c \)
For the quadratic in \( c \) to be positive for all \( c \), the discriminant of this quadratic must be less than or equal to zero:

\[ (4a - 2b)^2 - 4 \cdot 1 \cdot (b^2 - 4a^2 + 4ab) < 0 \]

### Step 5: Simplifying the condition
Expanding and simplifying:

\[ 16a^2 - 16ab + 4b^2 - 4b^2 + 16a^2 - 16ab < 0 \]

This simplifies to:

\[ 32a^2 - 32ab < 0 \]

Factoring out \( 16a \):

\[ 16a(2a - 2b) < 0 \]

### Step 6: Analyzing the inequality
This inequality implies:

1. \( a > 0 \) and \( a < b \) (which means \( b > a > 0 \))
2. \( a < 0 \) and \( a > b \) (which means \( b < a < 0 \))

### Conclusion
Thus, we have two cases:
- \( b > a > 0 \)
- \( b < a < 0 \)

The options provided in the question are:
- a. \( b < 0 < a \)
- b. \( a < 0 < b \)
- c. \( b < a < 0 \)
- d. \( b > a > 0 \)

From our analysis, the correct options that satisfy the conditions are:
- \( b < a < 0 \) (Option C)
- \( b > a > 0 \) (Option D)

### Final Answer
The conditions for the quadratic equation to have unequal real roots for all \( c \in \mathbb{R} \) are:
- \( b < a < 0 \) (Option C)
- \( b > a > 0 \) (Option D)