If the roots `ax^(2)+bx+c=0` are `(3)/(2),(4)/(3)` then `(a+b+c)^(2)` =

Options

A. `(a^(2))/(36)`
B. `(a^(2))/(6)`
C. `(a^(2))/(25)`
D. `(a^(2))/(5)`

Answer

To solve the problem, we need to find the value of \((a + b + c)^2\) given that the roots of the quadratic equation \(ax^2 + bx + c = 0\) are \(\frac{3}{2}\) and \(\frac{4}{3}\).

### Step-by-Step Solution:

1. **Identify the roots**: The roots of the quadratic equation are given as \(\alpha = \frac{3}{2}\) and \(\beta = \frac{4}{3}\).

2. **Use the sum of the roots**: According to Vieta's formulas, the sum of the roots \(\alpha + \beta\) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values of the roots:
\[ \frac{3}{2} + \frac{4}{3} = -\frac{b}{a} \]
To add these fractions, we find a common denominator:
\[ \frac{3}{2} = \frac{9}{6}, \quad \frac{4}{3} = \frac{8}{6} \]
Therefore,
\[ \frac{9}{6} + \frac{8}{6} = \frac{17}{6} \]
Thus, we have:
\[ -\frac{b}{a} = \frac{17}{6} \implies b = -\frac{17}{6}a \]

3. **Use the product of the roots**: The product of the roots \(\alpha \cdot \beta\) is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Substituting the values of the roots:
\[ \frac{3}{2} \cdot \frac{4}{3} = \frac{c}{a} \]
Simplifying the left side:
\[ \frac{3 \cdot 4}{2 \cdot 3} = \frac{4}{2} = 2 \]
Thus, we have:
\[ \frac{c}{a} = 2 \implies c = 2a \]

4. **Calculate \(a + b + c\)**: Now we can find \(a + b + c\):
\[ a + b + c = a + \left(-\frac{17}{6}a\right) + 2a \]
Combining the terms:
\[ a - \frac{17}{6}a + 2a = \left(1 + 2 - \frac{17}{6}\right)a = \left(3 - \frac{17}{6}\right)a \]
Converting \(3\) to sixths:
\[ 3 = \frac{18}{6} \implies \left(\frac{18}{6} - \frac{17}{6}\right)a = \frac{1}{6}a \]

5. **Calculate \((a + b + c)^2\)**:
\[ (a + b + c)^2 = \left(\frac{1}{6}a\right)^2 = \frac{1}{36}a^2 \]

### Final Result:
Thus, we find that:
\[ (a + b + c)^2 = \frac{a^2}{36} \]

### Conclusion:
The answer is \(\frac{a^2}{36}\).