Certainly! Let's solve this step-by-step.
Given:
[tex]\[ x = 3 + b i \][/tex]
[tex]\[ |x|^2 = 13 \][/tex]
Recall that the magnitude squared of a complex number [tex]\(a + bi\)[/tex] is given by:
[tex]\[ |a + bi|^2 = a^2 + b^2 \][/tex]
For our complex number [tex]\(x\)[/tex]:
[tex]\[ |3 + bi|^2 = 3^2 + b^2 \][/tex]
We are given that:
[tex]\[ |x|^2 = 13 \][/tex]
Substituting the values, we get:
[tex]\[ 3^2 + b^2 = 13 \][/tex]
Now, calculate [tex]\(3^2\)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
Substitute this back into the equation:
[tex]\[ 9 + b^2 = 13 \][/tex]
To isolate [tex]\(b^2\)[/tex], subtract 9 from both sides:
[tex]\[ b^2 = 13 - 9 \][/tex]
[tex]\[ b^2 = 4 \][/tex]
To find [tex]\(b\)[/tex], take the square root of both sides:
[tex]\[ b = \sqrt{4} \][/tex]
[tex]\[ b = 2 \quad \text{or} \quad b = -2 \][/tex]
Therefore, a possible value of [tex]\(b\)[/tex] is [tex]\(2\)[/tex] (since [tex]\(2\)[/tex] is listed in the provided options).
So, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
