Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Thus, \(\frac{x^{2}+11}{3-x}\) evaluates to \(-\frac{3}{5} - \frac{4}{5}i\) for \(x=4 i\).

Step 1: Substitute the value of x

Start by substituting \(x = 4i\) into the function \(\frac{x^{2}+11}{3-x}\), this gives \(\frac{(4i)^{2}+11}{3-(4i)} = \frac{-16+11}{3-4i}\), since \(i^{2} = -1\).

Step 2: Simplify the numerator

Simplify the numerator to get \(-5\) resulting to \(\frac{-5}{3-4i}\).

Step 3: Rationalize the denominator

Multiply both numerator and denominator by the conjugate of \(3-4i\), which is \(3+4i\). Now you have \(\frac{-5(3 + 4i)}{(3-4i)(3+4i)}\).

Step 4: Apply foil method to the denominator

Expand the denominator using FOIL: \(3*3 -3*4i +3*4i -(4i)^2\). This simplifies to \(9+16 = 25\), resulting in \(\frac{-5(3 + 4i)}{25}\).

Step 5: Simplify the expression

Finally, simplify the expression by distributing the -5 to both terms in the numerator and dividing by 25: \(\frac{-15 - 20i}{25} = -\frac{3}{5} - \frac{4}{5}i\).