Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(1+i\).

Step 2: Substitution

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

Step 3: Evaluate the Square

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

Step 4: Simplify

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

Step 5: Simplify Further

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