When (√5+3i)/(1+i) is expressed in the form a + bi, what is the value of b?

Rationale

(- 3 + √(5))/2

To express the complex fraction ((sqrt{5} + 3i)/(1 + i)) in the form (a + bi), we multiply the numerator and denominator by the conjugate of the denominator, yielding the correct form with (b) as ((-3 + sqrt{5})/2).

A (-1.5) YOUR ANSWER

This choice is a numerical value but does not represent the imaginary component of the expression. The imaginary part derived from the original complex fraction calculation does not simplify to a constant like -1.5, making it incorrect.

B (2) YOUR ANSWER

This value does not match the imaginary part obtained from simplifying the expression. After rationalizing the denominator and combining like terms, the imaginary component is (-3 + sqrt(5))/2 not 2. Hence, this option is incorrect.

C ((3-√5)/2) YOUR ANSWER

This choice incorrectly calculates the imaginary part of the expression. The manipulation of the complex fraction does not yield this value, as the correct imaginary component can be derived from the proper simplification of the expression.

D ((- 3 + √(5))/2) CORRECT

This option accurately represents the imaginary part of the expression after simplification. By rationalizing the denominator and combining terms, we find that the imaginary component indeed simplifies to ((-3 + sqrt{5})/2).

E ((- 3 - √(5))/2) YOUR ANSWER

This choice misrepresents the imaginary part of the fraction. The negative sign in front of (sqrt{5}) is incorrect based on the calculations performed during the simplification process, confirming that this is not a valid result.

Conclusion

In simplifying the complex fraction ((sqrt{5} + 3i)/(1 + i)), the correct imaginary component is ((-3 + sqrt{5})/2). This value reflects the proper manipulation of the expression, ensuring that other options do not represent valid outcomes from the operations performed. Thus, choice D emerges as the accurate representation of the imaginary part (b).

When (√5+3i)/(1+i) is expressed in the form a + bi, what is the value of b?

(- 3 + √(5))/2

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