/*! 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 51 Evaluate \(x^{2}-2 x+2\) for \(x... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 51

Evaluate \(x^{2}-2 x+2\) for \(x=1+i\)

Short Answer

Expert verified
The value of \(x^{2}-2 x+2\) for \(x=1+i\) is 0.

Step by step solution

01

Understand the Exercise

In the given exercise, the polynomial function \(x^{2}-2 x+2\) needs to be evaluated with \(x\) replaced by a complex number \(1+i\).
02

Substitution

Replace \(x\) with \(1+i\) in the expression \(x^{2}-2 x+2\). So the expression becomes \((1+i)^{2}-2(1+i)+2\).
03

Evaluate the Square

Calculate the square of \(1+i\). Using the formula \((a+b)^{2}=a^{2}+2ab+b^{2}\), it becomes \((1+i)^{2}=1^{2}+2*1*i+(i)^{2}=1+2i+(-1)= 2i\).
04

Simplify

Continue to simplify the expression by calculating \(2(1+i)\), resulting in \(2+2i\). So, the expression becomes \((1+i)^{2}-2(1+i)+2 = 2i-(2+2i)+2 = 2i-2-2i+2 \).
05

Simplify Further

Carry out the addition or subtraction from left to right, \(2i-2-2i+2 = 0\).

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