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Chapter 1: Problem 57

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

Short Answer

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

Step by step solution

01

Substituting x

Substitute \(x\) with the given value \(1+i\) in the equation. This will give us \((1+i)^2-2*(1+i)+2\).
02

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.
03

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 \(2-2 = 0\).

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Most popular questions from this chapter

In Exercises \(9-20,\) find each product and write the result in standard form. $$(2+3 i)^{2}$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x^{2}-3 x+2}{x^{2}-2 x-3}>0 $$

Explain how to add complex numbers. Provide an example with your explanation.

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(-3-\sqrt{-7})^{2}$$

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{5 i}{2-i}$$
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