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

Options

A. `|(b)/(c) - 4|ge 2sqrt(7)`
B. `|(c)/(b) - 4|ge 2sqrt(7)`
C. `|(a)/(c) - 11|ge 4sqrt(7)`
D. `|(a)/(b) + 11|ge 1sqrt((13)/(3))`

Explanation

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

### Step-by-Step Solution:

1. **Identify the Condition for Real Roots:**
The quadratic equation \( ax^2 + bx + c = 0 \) has real roots if its discriminant \( D \) is greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Therefore, we need:
\[ b^2 - 4ac \geq 0 \]

2. **Substitute \( a \) in the Discriminant:**
Since we know \( a = 2b + 3c \), we substitute this into the discriminant:
\[ b^2 - 4(2b + 3c)c \geq 0 \]
Simplifying this gives:
\[ b^2 - (8bc + 12c^2) \geq 0 \]
Rearranging, we have:
\[ b^2 - 8bc - 12c^2 \geq 0 \]

3. **Divide by \( c^2 \):**
To simplify the inequality, we can divide the entire inequality by \( c^2 \) (since \( c > 0 \)):
\[ \frac{b^2}{c^2} - \frac{8b}{c} - 12 \geq 0 \]
Let \( x = \frac{b}{c} \). Then the inequality becomes:
\[ x^2 - 8x - 12 \geq 0 \]

4. **Solve the Quadratic Inequality:**
We can solve the quadratic equation \( x^2 - 8x - 12 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{64 + 48}}{2} = \frac{8 \pm \sqrt{112}}{2} = \frac{8 \pm 4\sqrt{7}}{2} = 4 \pm 2\sqrt{7} \]
The roots are:
\[ x_1 = 4 + 2\sqrt{7}, \quad x_2 = 4 - 2\sqrt{7} \]

5. **Determine the Intervals:**
The quadratic \( x^2 - 8x - 12 \) opens upwards (since the coefficient of \( x^2 \) is positive). Therefore, the inequality \( x^2 - 8x - 12 \geq 0 \) holds for:
\[ x \leq 4 - 2\sqrt{7} \quad \text{or} \quad x \geq 4 + 2\sqrt{7} \]

6. **Substituting Back for \( b \):**
Recall that \( x = \frac{b}{c} \). Thus, we have:
\[ \frac{b}{c} \leq 4 - 2\sqrt{7} \quad \text{or} \quad \frac{b}{c} \geq 4 + 2\sqrt{7} \]
Since \( b \) and \( c \) are positive, we focus on:
\[ \frac{b}{c} \geq 4 + 2\sqrt{7} \]

### Conclusion:
The equation \( ax^2 + bx + c = 0 \) has real roots if:
\[ \frac{b}{c} \geq 4 + 2\sqrt{7} \]