Problem 57 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Substituting x

Substitute \(x\) with the given value \(1+i\) in the equation. This will give us \((1+i)^2-2*(1+i)+2\).

Step 2: Calculate the Square

Calculate the square of \(1+i\). Use the formula \((a+b)^2 = a^2 + 2ab + b^2\) to evaluate \((1+i)^2\) which results in \(1+2i+i^2\). But, \(i^2=-1\), substituting it in the equation gives us \(1+2i-1=2i\). The equation becomes \(2i - 2*(1+i) + 2\) after this calculation.

Step 3: Simplify the Equation

Simplify the equation by carrying out the operations. The term -2*(1+i) results in -2 - 2i. So, the equation becomes \(2i - 2 - 2i + 2\). Adding like terms simplifies the equation to \(2-2 = 0\).