Can a function really have no critical points?

AI Thread Summary
Complex numbers are introduced in Algebra 2 but often overlooked in subsequent courses to avoid confusion with real roots, especially when teaching quadratics. Educators prioritize real-world applications, leading to the dismissal of complex roots as "no solution" in practical problems. This approach aims to simplify learning for students who may struggle with foundational concepts. The discussion highlights the tension between theoretical mathematics and practical application in education. Understanding this context is essential for grasping the role of complex numbers in advanced mathematics.
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why is it that in algebra 2 people are told about complex numbers then they magically disappear for like 5 years and there not suppose to exist... sorry for wrong title I can't change it =(
 
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I don't quite understand what you're asking. What magically disappears? The complex numbers?

Say when you work with quadratics, the teachers don't want to start confusing you with complex roots when it's already enough strain on the student to understand everything there is to know about real roots. When real world applications are involved in the problems at hand, complex numbers are going to be ignored and just understood as "There is no solution".
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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