Given the expression \( z^2 + kz + 3 \), we want to find the values of \( k \) for which this expression can be expressed as a perfect square.
For the quadratic \( z^2 + kz + 3 \) to be a perfect square, it must equal \( (z + a)^2 \) for some real number \( a \). Expanding the perfect square gives us:
\[ (z + a)^2 = z^2 + 2az + a^2 \]
By comparing coefficients, we can match terms from both quadratic expressions:
- The coefficient of \( z \): \[ k = 2a \]
- The constant term: \[ 3 = a^2 \]
From the equation \( 3 = a^2 \), we have:
\[ a = \sqrt{3} \quad \text{or} \quad a = -\sqrt{3} \]
Using both values of \( a \), we can find \( k \):
- If \( a = \sqrt{3} \): \[ k = 2\sqrt{3} \]
- If \( a = -\sqrt{3} \): \[ k = -2\sqrt{3} \]
Thus, the possible values for \( k \) that make \( z^2 + kz + 3 \) a perfect square are:
\[ k = 2\sqrt{3} \quad \text{or} \quad k = -2\sqrt{3} \]