Suppose a,`b , c in R`. If the equations `ax^(2) + bx + c = 0 and 4x^(2) + 4x + 5 . 52` = 0 have a common root, then `(c)/(a)` is equal to ______

Answer

To solve the problem, we need to find the value of \( \frac{c}{a} \) given that the equations \( ax^2 + bx + c = 0 \) and \( 4x^2 + 4x + 5.52 = 0 \) have a common root.

### Step-by-step Solution:

1. **Identify the equations:**
- The first equation is \( ax^2 + bx + c = 0 \).
- The second equation is \( 4x^2 + 4x + 5.52 = 0 \).

2. **Calculate the discriminant of the second equation:**
- The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by \( D = B^2 - 4AC \).
- For the second equation, \( A = 4 \), \( B = 4 \), and \( C = 5.52 \).
- Thus, the discriminant is:
\[ D = 4^2 - 4 \cdot 4 \cdot 5.52 = 16 - 88.32 = -72.32 \]
- Since the discriminant is negative, the roots are complex and occur in conjugate pairs.

3. **Let the common root be \( \alpha + i\beta \) and \( \alpha - i\beta \):**
- Both equations share these roots.

4. **Using the product of the roots:**
- The product of the roots of the second equation can be calculated as:
\[ \text{Product of roots} = \frac{C}{A} = \frac{5.52}{4} = 1.38 \]

5. **For the first equation \( ax^2 + bx + c = 0 \):**
- The product of the roots is also given by \( \frac{c}{a} \).

6. **Setting the two expressions for the product of the roots equal:**
- Since both equations have the same product of roots, we have:
\[ \frac{c}{a} = 1.38 \]

7. **Conclusion:**
- Therefore, the value of \( \frac{c}{a} \) is \( 1.38 \).

### Final Answer:
\[ \frac{c}{a} = 1.38 \]