To find the magnitude (or modulus) of a complex number [tex]\( z = a + bi \)[/tex], we use the formula:
[tex]\[
|z| = \sqrt{a^2 + b^2}
\][/tex]
Given the complex number [tex]\( z = -3 + 2i \)[/tex]:
1. Identify the real part [tex]\( a \)[/tex] and the imaginary part [tex]\( b \)[/tex]:
[tex]\[
a = -3, \quad b = 2
\][/tex]
2. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the magnitude formula:
[tex]\[
|z| = \sqrt{(-3)^2 + (2)^2}
\][/tex]
3. Calculate [tex]\( (-3)^2 \)[/tex] and [tex]\( 2^2 \)[/tex]:
[tex]\[
(-3)^2 = 9, \quad 2^2 = 4
\][/tex]
4. Add these results together:
[tex]\[
9 + 4 = 13
\][/tex]
5. Take the square root of the sum:
[tex]\[
|z| = \sqrt{13}
\][/tex]
Therefore, the magnitude [tex]\( |z| \)[/tex] of the complex number [tex]\( z = -3 + 2i \)[/tex] is [tex]\( \sqrt{13} \)[/tex].
The correct answer is:
[tex]\[
\sqrt{13}
\][/tex]
