Problem 54 - College Algebra Essentials

Question

Evaluate \(\frac{x^{2}+11}{3-x}\) for \(x=4 i\)

Step-by-Step Solution

Answer

The result of evaluating the expression \(\frac{x^{2}+11}{3-x}\) at \(x=4i\) is \(-\frac{3}{5}-\frac{4i}{5}\).

Step 1: Substitute the value of x

Begin by replacing the variable \(x\) in the given expression \(\frac{x^{2}+11}{3-x}\) with \(4i\), resulting in: \(\frac{(4i)^{2}+11}{3-(4i)}\).

Step 2: Simplify the numerator

Simplify the numerator \((4i)^{2}+11\). The squaring of \(4i\) results in \((-16)+11 = -5\). So the numerator is \(-5\).

Step 3: Simplify the denominator

Simplify the denominator \(3-(4i)\). So the denominator is \(3-4i\).

Step 4: Writing the final expression

Integrate the results from step 2 and 3, giving us \(\frac{-5}{3-4i}\). To make the denominator a real number, we'll multiply the fraction by the conjugate of the denominator \((3+4i)\), resulting in \(\frac{-5(3+4i)}{(3-4i)(3+4i)}\). After simplifying the expression, it results in: \(\frac{-15-20i}{25}\). Dividing each term in the numerator by 25 gives us the final solution \(-\frac{3}{5}-\frac{4i}{5}\).