Question

Explain why $2-i$ is a solution to$$x^{2}-4 x+5=0$$

   Explain why $2-i$ is a solution to$$x^{2}-4 x+5=0$$

Algebra for College Students

Algebra for College Students

Mark Dugopolski 5th Edition

Chapter 7, Problem 109

Instant Answer

Solved by Expert Heather Zimmers

04/14/2020

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Step 1: Substitute $2-i$ for $x$ in the given equation: $$(2-i)^{2}-4(2-i)+5=0$$ Show more…

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Explain why $2-i$ is a solution to$$x^{2}-4 x+5=0$$

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Key Concepts

Complex Numbers

Complex numbers extend the real numbers and are written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part, and i represents the square root of -1. They allow for solutions to equations that do not have real solutions, ensuring that every polynomial equation has the necessary number of roots according to its degree.

Verification by Substitution

This is a method used to determine whether a particular value is a solution of an equation. By substituting the value into the equation and simplifying, one can check if the equation holds true (i.e., the expression equals zero), thereby confirming that the candidate is indeed a valid solution.

Complex Conjugate Root Theorem

For any polynomial with real coefficients, if a complex number is a solution, then its complex conjugate is also guaranteed to be a solution. This theorem ensures that non-real roots appear in conjugate pairs, which maintains the real nature of the polynomial's coefficients.

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