If `a gt 2`, then the roots of the equation `(2-a)x^(2)+3ax-1=0` are

Options

A. one positive and one negative
B. both negative
C. both positive
D. both imaginary

Answer

To determine the nature of the roots of the equation \((2-a)x^2 + 3ax - 1 = 0\) given that \(a > 2\), we will analyze the coefficients and apply the properties of quadratic equations.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The equation can be rewritten in the standard form \(Ax^2 + Bx + C = 0\), where:
- \(A = 2 - a\)
- \(B = 3a\)
- \(C = -1\)

2. **Determine the sign of \(A\)**:
Since \(a > 2\), we have:
\[ 2 - a < 0 \quad \text{(because \(a\) is greater than 2)} \]
Therefore, \(A < 0\).

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 \(C\):
\[ D = (3a)^2 - 4(2-a)(-1) \]
Simplifying this:
\[ D = 9a^2 + 4(2-a) = 9a^2 + 8 - 4a \]
\[ D = 9a^2 - 4a + 8 \]

4. **Analyze the discriminant**:
To determine the nature of the roots, we need to check if \(D\) is positive, negative, or zero. We can analyze the quadratic \(9a^2 - 4a + 8\):
- The discriminant of this quadratic (in terms of \(a\)) is:
\[ (-4)^2 - 4 \cdot 9 \cdot 8 = 16 - 288 = -272 \]
Since the discriminant is negative, \(9a^2 - 4a + 8\) has no real roots and is always positive for all real \(a\).

5. **Conclusion about the roots**:
Since \(D > 0\) (as \(9a^2 - 4a + 8 > 0\)), the quadratic equation has two distinct real roots. However, since \(A < 0\), the parabola opens downwards. Therefore, both roots must be positive.

### Final Answer:
The roots of the equation \((2-a)x^2 + 3ax - 1 = 0\) are both positive.