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Chapter 1: Problem 51

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

Short Answer

Expert verified
Evaluating \(x^{2}-2 x+2\) for \(x=1+i\), the result is 0.

Step by step solution

01

Substitution

First, replace \(x\) in the given polynomial \(x^{2}-2 x+2\) by the complex number \(1+i\). This leads to: \((1+i)^{2}-2(1 + i)+2\).
02

Calculation of the Squared Term

Next, compute the square of \(1+i\). Using the formula \((a+b)^2 = a^2 + 2ab + b^2\), this gives: \((1+i)^{2}=1^2 + 2*1*i + (i)^2 = 1+2i -1 = 2i\). Therefore, our expression transforms to: \((1+i)^{2}=2i\). Substitute this into the original equation to obtain: \(2i -2(1 + i)+2\).
03

Simplification

Now let's simplify the expression. Calculate the second term, which is the double of \(1 + i\) : \(-2(1 + i) = -2 - 2i\). The polynomial now becomes: \(2i -2 -2i +2\). It can be seen that \(2i\) and \(-2i\) cancel each other leaving us with an equation: \(-2 + 2\).
04

Final Simplification

Finally, simplify the equaition \(-2 + 2\). It simplifies to 0.

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