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Chapter 2: Problem 57

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

Short Answer

Expert verified
The evaluated expression \(x^{2} - 2x + 2\) for \(x = 1 + i\) equals -1.

Step by step solution

01

Replace x with the given value

Start by replacing the variable \(x\) in the expression with the complex number \(x = 1 + i\). This gives us \((1+i)^{2}-2(1 + i)+2\).
02

Simplify the Expression

Expand \((1+i)^{2}\) by squaring both real and imaginary part and remember that \(i^2 = -1\). Then calculate the remaining terms. This gives us \(1 + 2i + (i^2)- 2-2i +2 = 1+2i-1-2-2i+2 = -1\).
03

Solution

Finally, note that the imaginary part sums to zero, and the real parts combine to give our solution of -1.

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