Questions: Evaluate (x^2+11)/(5-x) for x=2i (x^2+11)/(5-x) evaluated for x=2i equals . (Type your answer in the form a + bi. Use integers or simplified fractions for any numbers in the expression.)

$Evaluate (x^2+11)/(5-x) for x=2i (x^2+11)/(5-x) evaluated for x=2i equals . (Type your answer in the form a + bi. Use integers or simplified fractions for any numbers in the expression.)$

Transcript text: Evaluate $\frac{x^{2}+11}{5-x}$ for $x=2 i$ $\frac{x^{2}+11}{5-x}$ evaluated for $x=2 i$ equals $\square$ . (Type your answer in the form a + biUse integers or simplified fractions for any numbers in the expression.)

Solution

Solution Steps

To evaluate the expression $\frac{x^{2}+11}{5-x}$ for $x=2i$, substitute $x=2i$ into the expression. Simplify the numerator and the denominator separately, then divide the results to find the complex number in the form $a + bi$.

Step 1: Substitute $x = 2i$ into the Expression

To evaluate the expression $\frac{x^2 + 11}{5 - x}$ for $x = 2i$, substitute $x = 2i$ into the expression:

\[ \frac{(2i)^2 + 11}{5 - 2i} \]

Step 2: Simplify the Numerator

Calculate the numerator:

\[ (2i)^2 + 11 = 4i^2 + 11 = 4(-1) + 11 = -4 + 11 = 7 \]

Step 3: Simplify the Denominator

The denominator is:

\[ 5 - 2i \]

Step 4: Rationalize the Denominator

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{7}{5 - 2i} \times \frac{5 + 2i}{5 + 2i} = \frac{7(5 + 2i)}{(5 - 2i)(5 + 2i)} \]