/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Evaluate \(\frac{x^{2}+19}{2-x}\... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 53

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

Short Answer

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

Step by step solution

01

Substitution

Substitute \(x = 3i\) into the function \(f(x) = \frac{x^{2}+19}{2-x}\). This gives \(f(3i) = \frac{(3i)^{2}+19}{2-3i}\)
02

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: \(2 - 3i\)
03

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: \( \frac{10(2 + 3i)}{(2 - 3i)(2 + 3i)}\).
04

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, \( \frac{10(2 + 3i)}{13} = \frac{20}{13} + \frac{30i}{13}.\)

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