Problem 52 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution to the expression \(x^{2} - 2x + 5\) when evaluated for \(x = 1-2i\) is 0.

Step 1: Substituting the value of x

First, we substitute \(x = 1 - 2i\) into the given polynomial \(x^{2}-2 x+5\) to get \((1 - 2i)^{2} - 2*(1 - 2i) + 5\).

Step 2: Simplifying the Expression

To simplify this, start by expanding the first term \((1 - 2i)^{2}\), square each term and then cross multiply. The result is \(1 - 4i + 4i^{2}\). The second term becomes \(-2 + 4i\). Finally, add 5. The expression then becomes \(1 - 4i + 4i^{2} - 2 + 4i + 5\).

Step 3: Continue to Simplify Using \(i^{2}=-1\)

We know that \(i^{2}=-1\). With this, the above expression becomes \(1 - 4i + 4*(-1) - 2 + 4i + 5 = 1 - 2 + 5 - 4\).