To find [tex]\( w \)[/tex] from the given complex numbers [tex]\( z_1 = 2 - i \)[/tex] and [tex]\( z_2 = -2 + i \)[/tex], we divide [tex]\( z_1 \)[/tex] by [tex]\( z_2 \)[/tex]:
[tex]\[ w = \frac{z_1}{z_2} = \frac{2 - i}{-2 + i} \][/tex]
To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(-2 + i\)[/tex] is [tex]\(-2 - i\)[/tex]:
[tex]\[ w = \frac{(2 - i)(-2 - i)}{(-2 + i)(-2 - i)} \][/tex]
First, we calculate the denominator:
[tex]\[ (-2 + i)(-2 - i) = (-2)^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \][/tex]
Next, we calculate the numerator:
[tex]\[ (2 - i)(-2 - i) = 2(-2) + 2(-i) - i(-2) - i(i) \][/tex]
[tex]\[ = -4 - 2i + 2i - i^2 \][/tex]
[tex]\[ = -4 - i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ -4 - (-1) = -4 + 1 = -3 \][/tex]
Now, we have:
[tex]\[ w = \frac{-3}{5} = -\frac{3}{5} \][/tex]
This result is purely real with no imaginary part, so:
[tex]\[ w = -1 - 0i \][/tex]
Thus, if we write [tex]\( w \)[/tex] in the form [tex]\( w = a + bi \)[/tex], we have:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = 0 \][/tex]
Therefore, the complex number [tex]\( w \)[/tex] in the form [tex]\( a + bi \)[/tex] is:
[tex]\[ w = -1 + 0i \][/tex]
