Problem 52 - College Algebra Essentials

Question

Evaluate \(x^{2}-2 x+5\) for \(x=1-2 i\)

Step-by-Step Solution

Answer

The value of \(x^{2}-2 x+5\) when \(x=1-2i\) is 0.

Step 1: Substitute \(x\)

First, substitute \(x=1-2i\), into the equation \(x^{2}-2 x+5\). We have \((1-2i)^{2}-2(1-2i)+5\).

Step 2: Resolve Square

Expand the expression \((1-2i)^{2}\). Remembering that \(i^{2}=-1\), it's \(1^2 - 2*1*2i + (2i)^{2} = 1 - 4i - 4\). So the expression now reads as \((-3 - 4i) - 2(1-2i) + 5\).

Step 3: Continue to Simplify

Continue to simplify the expression: \(-3 - 4i - 2 + 4i + 5\). The \(4i\) and \(-4i\) will cancel each other, and we are left with \(-3 - 2 + 5 = 0\).