Problem 51 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substitution

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

Step 2: 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: \(2i -2(1 + i)+2\).

Step 3: 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: \(-2 + 2\).

Step 4: Final Simplification

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