To find the possible value or values of [tex]\( z \)[/tex] in the quadratic equation [tex]\( z^2 - 4z + 4 = 0 \)[/tex], we follow these steps:
1. Identify the quadratic equation: [tex]\( z^2 - 4z + 4 = 0 \)[/tex].
2. Factor the equation: We can rewrite the quadratic equation in its factored form. We look for two numbers that multiply to give the constant term (4) and add to give the coefficient of [tex]\( z \)[/tex] (-4). Observing the equation:
[tex]\[
z^2 - 4z + 4 = (z - 2)(z - 2) = (z - 2)^2 = 0
\][/tex]
3. Solve the factored equation: Set each factor equal to zero.
[tex]\[
(z - 2)^2 = 0
\][/tex]
This implies:
[tex]\[
z - 2 = 0
\][/tex]
4. Find the solution for [tex]\( z \)[/tex]:
[tex]\[
z = 2
\][/tex]
After solving the equation, we see that the quadratic equation [tex]\( z^2 - 4z + 4 = 0 \)[/tex] has a repeated root, [tex]\( z = 2 \)[/tex].
Hence, the best answer from the given options is:
D. [tex]\( z = 2 \)[/tex].
