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12. Find the possible value or values of [tex][tex]$z$[/tex][/tex] in the quadratic equation [tex][tex]$z^2 - 4z + 4 = 0$[/tex][/tex].

A. [tex][tex]$z = \sqrt{2} + 2, z = -\sqrt{2} + 2$[/tex][/tex]
B. [tex][tex]$z = 2\sqrt{2} + 2, z = -\sqrt{2} + 2$[/tex][/tex]
C. [tex][tex]$z = 10, z = 6$[/tex][/tex]
D. [tex][tex]$z = 2$[/tex][/tex]

Sagot :

GinnyAnswer
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].
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