Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

If the complex number [tex]$x=3+bi$[/tex] and [tex]|x|^2=13[/tex], which is a possible value of [tex]b[/tex]?

A. 2
B. 4
C. 9
D. 10


Sagot :

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]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.