Problem 53 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

Evaluating the given expression for \(x=3i\), we get: \(f(3i) = \frac{20}{13} + \frac{30i}{13}\)

Step 1: Substitution

Substitute \(x = 3i\) into the function \(f(x) = \frac{x^{2}+19}{2-x}\). This gives \(f(3i) = \frac{(3i)^{2}+19}{2-3i}\)

Step 2: Simplify Numerator and Denominator Separately

Simplify the expression by first dealing with the numerator and the denominator separately. In the numerator, calculate the square of \(3i\) and add 19. This gives: \((3i)^{2} + 19 = -9 + 19 = 10\). In the denominator, subtract \(3i\) from 2. This gives: \(2 - 3i\)

Step 3: Rationalize the Denominator

Here we deal with the denominator: we need to get rid of the imaginary number 'i' in the denominator. Multiply both the numerator and the denominator by the conjugate of the denominator \(2 + 3i\). This gives: \( \frac{10(2 + 3i)}{(2 - 3i)(2 + 3i)}\).

Step 4: Simplify the Result

In the denominator, we now have the difference of squares which simplifies to \(2^{2} - (3i)^{2} = 4 - (-9) = 13\). Now, in the numerator calculate the product: \(10(2 + 3i) = 20 + 30i\). Therefore, \( \frac{10(2 + 3i)}{13} = \frac{20}{13} + \frac{30i}{13}.\)