If a, b, c are positive and a = 2b + 3c, then roots of the equation `ax^(2) + bx + c = 0` are real for

Options

A. `|(a)/(c)-11| ge 4 sqrt(7)`
B. `|(c)/(a)-11| ge 4 sqrt(7)`
C. `|(b)/(c)+4| ge 2 sqrt(7)`
D. `|(c)/(b)-4| ge 2 sqrt(7)`

Explanation

To solve the problem, we need to determine the conditions under which the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real, given that \( a = 2b + 3c \) and \( a, b, c \) are positive.

### Step-by-Step Solution:

1. **Identify the Discriminant**:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real if the discriminant \( D \) is greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]

2. **Substitute \( a \)**:
We know that \( a = 2b + 3c \). We will substitute this into the discriminant:
\[ D = b^2 - 4(2b + 3c)c \]
Simplifying this gives:
\[ D = b^2 - 8bc - 12c^2 \]

3. **Set the Discriminant Greater Than or Equal to Zero**:
For the roots to be real, we need:
\[ b^2 - 8bc - 12c^2 \geq 0 \]

4. **Rearranging the Inequality**:
Rearranging the inequality gives:
\[ b^2 - 8bc - 12c^2 \geq 0 \]

5. **Using the Quadratic Formula**:
We can treat this as a quadratic in \( b \):
\[ b^2 - 8bc - 12c^2 = 0 \]
The roots of this quadratic can be found using the quadratic formula:
\[ b = \frac{-(-8c) \pm \sqrt{(-8c)^2 - 4(1)(-12c^2)}}{2(1)} \]
Simplifying gives:
\[ b = \frac{8c \pm \sqrt{64c^2 + 48c^2}}{2} = \frac{8c \pm \sqrt{112c^2}}{2} \]
\[ b = \frac{8c \pm 4\sqrt{7}c}{2} = 4c \pm 2\sqrt{7}c \]

6. **Finding the Conditions**:
The roots are \( b = (4 + 2\sqrt{7})c \) and \( b = (4 - 2\sqrt{7})c \). Since \( b \) must be positive, we consider:
\[ b \geq (4 + 2\sqrt{7})c \quad \text{or} \quad b \leq (4 - 2\sqrt{7})c \]
However, since \( 4 - 2\sqrt{7} < 0 \), we discard that option. Thus, we have:
\[ b \geq (4 + 2\sqrt{7})c \]

### Final Condition:
Thus, the condition for the roots of the equation \( ax^2 + bx + c = 0 \) to be real is:
\[ b \geq (4 + 2\sqrt{7})c \]