The real values of a for which the quadratic equation `2x^2-(a^(3)+8a-1)x+a^(2)-4a=0` possesses roots of opposite signs are given by :

Options

A. `a gt6`
B. `a gt 9`
C. `0 lt a lt4`
D. `a lt 0`

Answer

To determine the real values of \( a \) for which the quadratic equation

\[ 2x^2 - (a^3 + 8a - 1)x + (a^2 - 4a) = 0 \]

has roots of opposite signs, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be expressed in the standard form \( Ax^2 + Bx + C = 0 \), where:
- \( A = 2 \)
- \( B = -(a^3 + 8a - 1) \)
- \( C = a^2 - 4a \)

### Step 2: Use the condition for roots of opposite signs
For a quadratic equation to have roots of opposite signs, the product of the roots must be negative. The product of the roots \( \alpha \) and \( \beta \) can be given by:

\[ \alpha \beta = \frac{C}{A} = \frac{a^2 - 4a}{2} \]

We need this product to be less than zero:

\[ \frac{a^2 - 4a}{2} < 0 \]

### Step 3: Simplify the inequality
Multiplying both sides by 2 (which is positive and does not change the inequality):

\[ a^2 - 4a < 0 \]

### Step 4: Factor the quadratic expression
Factoring the left-hand side gives:

\[ a(a - 4) < 0 \]

### Step 5: Determine the intervals
To find the intervals where this inequality holds, we identify the roots of the equation \( a(a - 4) = 0 \), which are \( a = 0 \) and \( a = 4 \).

Now, we test the intervals:
1. For \( a < 0 \): Choose \( a = -1 \) → \( (-1)(-1 - 4) = 5 > 0 \) (not valid)
2. For \( 0 < a < 4 \): Choose \( a = 2 \) → \( (2)(2 - 4) = -4 < 0 \) (valid)
3. For \( a > 4 \): Choose \( a = 5 \) → \( (5)(5 - 4) = 5 > 0 \) (not valid)

### Step 6: Conclusion
The valid interval for \( a \) where the roots of the quadratic equation are of opposite signs is:

\[ 0 < a < 4 \]

Thus, the real values of \( a \) for which the quadratic equation has roots of opposite signs are given by:

\[ \text{Answer: } a \in (0, 4) \]