Question

If the complex number x = 3 + bi and |x|² = 13, which is a possible value of b?

a. 2
b. 4
c. 9
d. 10

Answer 1

The possible value of b for the complex number x = 3 + bi when |x|² = 13 is found by solving the equation 3² + b² = 13. Subtracting 9 from both sides yields b² = 4, so the possible values of b are ±2, and thus 2 is a correct answer.

Step-by-step explanation:

To find the possible value of b for the complex number x = 3 + bi when |x|² = 13, we use the formula for the modulus of a complex number. The modulus squared, which is the sum of the squares of the real part and the imaginary part, is given as |x|² = (3)² + (b)². Plugging in the given modulus value and solving for b, we have:

3² + b² = 13

9 + b² = 13

b² = 13 - 9

b² = 4

b = ±2

Since b must be a real number, the possible values of b are 2 and -2. Therefore, the correct answer is a. 2.

Given the complex number x = 3 + bi and |x|² = 13, we can find the value of b by substituting the values into the formula: -13 ± √(13)² - 4 × 3 × (-10) / (2 × 3).

Simplifying the equation, we get: -13 ± √(169 - 120) / 6.

Using the quadratic formula, we can find the two possible values of b as 2 or -4. Therefore, the correct answer is (a) 2.