Question
If z square+kz+3 of 8 equal parts is a perfect square then k is?
Answers
GPT-4o mini
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:
1. The coefficient of \( z \):
\[
k = 2a
\]
2. 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 \( k \):
1. If \( a = \sqrt{3} \):
\[
k = 2\sqrt{3}
\]
2. If \( a = -\sqrt{3} \):
\[
k = -2\sqrt{3}
\]
Thus, the possible values for \( k \) that make \( z^2 + kz + 3 \) a perfect square are:
\[
k = 2\sqrt{3} \quad \text{or} \quad k = -2\sqrt{3}
\]
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:
1. The coefficient of \( z \):
\[
k = 2a
\]
2. 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 \( k \):
1. If \( a = \sqrt{3} \):
\[
k = 2\sqrt{3}
\]
2. If \( a = -\sqrt{3} \):
\[
k = -2\sqrt{3}
\]
Thus, the possible values for \( k \) that make \( z^2 + kz + 3 \) a perfect square are:
\[
k = 2\sqrt{3} \quad \text{or} \quad k = -2\sqrt{3}
\]