If both roots are less than 3 then 'a' belongs to

Options

A. `((3)/(2),3)`
B. `(-1,2)cup(2,3)`
C. `(-infty,2)`
D. `(2,infty)`

Answer

To solve the problem, we need to analyze the given quadratic equation and determine the conditions under which both roots are less than 3.

Let's denote the quadratic equation as:

\[ ax^2 + bx + c = 0 \]

### Step 1: Identify the conditions for the roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the roots can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

To ensure that both roots are less than 3, we need to satisfy two conditions:

1. The sum of the roots \( \frac{-b}{a} < 6 \) (since \( 3 + 3 = 6 \))
2. The product of the roots \( \frac{c}{a} < 9 \) (since \( 3 \times 3 = 9 \))

### Step 2: Apply the conditions to the coefficients
Let’s denote the coefficients of the quadratic equation as follows:

- \( a = 1 \)
- \( b = -6 \) (from the sum condition)
- \( c = 9 \) (from the product condition)

### Step 3: Set up the inequalities
From the conditions derived, we can set up the following inequalities:

1. \( -\frac{b}{a} < 6 \) implies \( -b < 6a \)
2. \( \frac{c}{a} < 9 \) implies \( c < 9a \)

### Step 4: Solve the inequalities
Substituting the values of \( a \), \( b \), and \( c \):

1. For \( -b < 6a \):
- If \( b < 0 \), then \( -b > 0 \) and the inequality holds.

2. For \( c < 9a \):
- If \( c < 9 \), then this condition is satisfied.

### Step 5: Conclusion
Thus, for both roots to be less than 3, we find that \( a \) must belong to a specific range.

### Final Answer
The values of \( a \) that satisfy both conditions are those for which the inequalities hold true.