Problem 53 - Algebra and Trigonometry

Question

Evaluate \(\frac{x^{2}+19}{2-x}\) for \(x=3 i\)

Step-by-Step Solution

Answer

The result of evaluating \(\frac{x^{2}+19}{2-x}\) for \(x=3 i\) is \( \frac{20}{13} + \frac{30i}{13} \).

Step 1: Substitute the value into the function

Replace \(x\) in the function \(\frac{x^{2}+19}{2-x}\) with \(3i\). This gives \(\frac{(3i)^{2}+19}{2-(3i)}\).

Step 2: Simplify the numerator and the denominator

Calculate \((3i)^{2}\) and \(2-(3i)\). The square of \(3i\) is \(-9\) (as \(i^{2}=-1\)), and the subtraction gives \(2-3i\). This leads to \(\frac{-9+19}{2-3i}\) or \(\frac{10}{2-3i}\).

Step 3: Rationalize the denominator

Multiply the numerator and the denominator by the conjugate of the denominator which is \(2+3i\). This gives \(\frac{10(2+3i)}{(2-3i)(2+3i)}\).

Step 4: Simplify the expression

Carry out the multiplication in both numerator and denominator. The denominator simplifies to 4 + 9 which equals 13. The numerator is \(20+30i\). So the expression simplifies to \(\frac{20+30i}{13}\). So this gives \( \frac{20}{13}+\frac{30i}{13} \).