If `(3+4i)/(2-4i)=x+iy`, then x=. . . . . And y=. . . .

Answer

To solve the equation \((3 + 4i)/(2 - 4i) = x + iy\), we need to find the values of \(x\) and \(y\). We will do this by rationalizing the denominator.

### Step-by-step Solution:

1. **Start with the given expression**:
\[ \frac{3 + 4i}{2 - 4i} \]

2. **Rationalize the denominator**:
To eliminate the imaginary part in the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is \(2 + 4i\):
\[ \frac{(3 + 4i)(2 + 4i)}{(2 - 4i)(2 + 4i)} \]

3. **Calculate the denominator**:
Using the difference of squares formula \(a^2 - b^2\):
\[ (2 - 4i)(2 + 4i) = 2^2 - (4i)^2 = 4 - 16(-1) = 4 + 16 = 20 \]

4. **Calculate the numerator**:
Distributing in the numerator:
\[ (3 + 4i)(2 + 4i) = 3 \cdot 2 + 3 \cdot 4i + 4i \cdot 2 + 4i \cdot 4i = 6 + 12i + 8i + 16i^2 \]
Since \(i^2 = -1\), we have:
\[ 16i^2 = 16(-1) = -16 \]
Thus, the numerator simplifies to:
\[ 6 + 20i - 16 = -10 + 20i \]

5. **Combine the results**:
Now we can rewrite the expression:
\[ \frac{-10 + 20i}{20} \]

6. **Separate into real and imaginary parts**:
This simplifies to:
\[ -\frac{10}{20} + \frac{20}{20}i = -\frac{1}{2} + i \]

7. **Identify \(x\) and \(y\)**:
From the expression \(-\frac{1}{2} + i\), we can identify:
\[ x = -\frac{1}{2}, \quad y = 1 \]

### Final Answer:
\[ x = -\frac{1}{2}, \quad y = 1 \]