If `(a + ib)^2 = x + iy` find `x^2 + y^2`

Answer

To solve the problem, we start with the equation given:

\[ (a + ib)^2 = x + iy \]

### Step 1: Expand the left-hand side
Using the identity for the square of a binomial, we can expand \((a + ib)^2\):

\[ (a + ib)^2 = a^2 + 2ab(i) + (ib)^2 \]

### Step 2: Substitute \(i^2\)
Since \(i^2 = -1\), we can substitute this into our equation:

\[ (a + ib)^2 = a^2 + 2abi - b^2 \]

This can be rearranged to separate the real and imaginary parts:

\[ = (a^2 - b^2) + (2ab)i \]

### Step 3: Equate real and imaginary parts
Now, we can equate the real and imaginary parts from both sides of the equation:

\[ x = a^2 - b^2 \]
\[ y = 2ab \]

### Step 4: Find \(x^2 + y^2\)
Next, we need to calculate \(x^2 + y^2\):

\[ x^2 + y^2 = (a^2 - b^2)^2 + (2ab)^2 \]

### Step 5: Expand \(x^2\) and \(y^2\)
Now, we expand both terms:

1. For \(x^2\):
\[ (a^2 - b^2)^2 = a^4 - 2a^2b^2 + b^4 \]

2. For \(y^2\):
\[ (2ab)^2 = 4a^2b^2 \]

### Step 6: Combine the results
Now, we combine the results of \(x^2\) and \(y^2\):

\[ x^2 + y^2 = (a^4 - 2a^2b^2 + b^4) + 4a^2b^2 \]

This simplifies to:

\[ x^2 + y^2 = a^4 + b^4 + 2a^2b^2 \]

### Step 7: Factor the expression
Notice that \(a^4 + b^4 + 2a^2b^2\) can be factored as:

\[ x^2 + y^2 = (a^2 + b^2)^2 \]

### Final Answer
Thus, the final result is:

\[ x^2 + y^2 = (a^2 + b^2)^2 \]

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