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

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

### Step 1: Identify the coefficients
The coefficients of the quadratic equation can be identified as follows:
- \( A = 2 - a \) (coefficient of \( x^2 \))
- \( B = 3a \) (coefficient of \( x \))
- \( C = -1 \) (constant term)

### Step 2: Determine the nature of the roots
To determine the nature of the roots of a quadratic equation, we can use the discriminant \( D \), which is given by:

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

Substituting the coefficients into the discriminant formula, we get:

\[ D = (3a)^2 - 4(2 - a)(-1) \]

### Step 3: Simplify the discriminant
Now, let's simplify the expression for \( D \):

\[ D = 9a^2 + 4(2 - a) \]
\[ D = 9a^2 + 8 - 4a \]
\[ D = 9a^2 - 4a + 8 \]

### Step 4: Analyze the discriminant
Next, we need to check the sign of the discriminant \( D \). Since \( a > 2 \), we can analyze the quadratic \( 9a^2 - 4a + 8 \).

The discriminant of this new quadratic can be calculated as follows:

\[ D' = (-4)^2 - 4 \cdot 9 \cdot 8 \]
\[ D' = 16 - 288 = -272 \]

Since the discriminant \( D' < 0 \), the quadratic \( 9a^2 - 4a + 8 \) does not have real roots, which means it is always positive for all \( a \).

### Step 5: Conclusion about the roots
Since \( D > 0 \) for \( a > 2 \), it indicates that the original quadratic equation has two distinct real roots.

### Final Answer
Thus, if \( a > 2 \), the roots of the equation \( (2 - a)x^2 + 3ax - 1 = 0 \) are **real and distinct**.

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