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

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(8)`
D. `(1)/(16)`

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 \(2\) and \(\frac{3}{2}\).

### Step-by-step Solution:

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

2. **Form the quadratic equation**: Using the roots, we can express the quadratic equation in factored form:
\[ (x - 2)\left(x - \frac{3}{2}\right) = 0 \]

3. **Expand the equation**: We will expand the product:
\[ (x - 2)\left(x - \frac{3}{2}\right) = x^2 - \frac{3}{2}x - 2x + 3 \]
Combining the \(x\) terms:
\[ x^2 - \left(2 + \frac{3}{2}\right)x + 3 = x^2 - \frac{4}{2}x - \frac{3}{2}x + 3 = x^2 - \frac{7}{2}x + 3 \]

4. **Set the equation equal to \(ax^2 + bx + c\)**: We can now compare coefficients:
\[ ax^2 + bx + c = x^2 - \frac{7}{2}x + 3 \]
From this, we can identify:
- \(a = 1\)
- \(b = -\frac{7}{2}\)
- \(c = 3\)

5. **Calculate \(a + b + c\)**:
\[ a + b + c = 1 - \frac{7}{2} + 3 \]
To simplify, convert \(1\) and \(3\) to fractions with a denominator of \(2\):
\[ 1 = \frac{2}{2}, \quad 3 = \frac{6}{2} \]
Thus,
\[ a + b + c = \frac{2}{2} - \frac{7}{2} + \frac{6}{2} = \frac{2 - 7 + 6}{2} = \frac{1}{2} \]

6. **Calculate \((a + b + c)^2\)**:
\[ (a + b + c)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Final Answer:
\[ \boxed{\frac{1}{4}} \]