Mathematics: Completing the square
Mathematics from high school to university
What you will learn
Completing the square: how it is done and why the method works (variants for 1, 2, and 3 variables); a geometric illustration of the method and its name.
Completing the square in one variable for solving quadratic equations with help of the discriminant and the quadratic formula; derivation of the formula.
Completing the square in one variable for plotting polynomials of second degree with help of graph transformations of the parabola y=x^2; finding new vertex.
Completing the square in two variables, one at a time, for identifying conic sections: circles, ellipses, parabolas, and hyperbolas.
Completing the square in three variables, one at a time, for identifying quadric surfaces: spheres, ellipsoids, hyperboloids, double cones, etc.
Completing the square in two or three variables for definiteness of 2x2 or 3x3 matrices / corresponding quadratic forms in 2 or 3 variables.
Why take this course?
๐งฎ Mathematics: Completing the Square ๐
Mathematics from High School to University
Understanding Completing the Square
Dive into the world of mathematics with our comprehensive course on "Compleating the Square." This essential technique is not just a high school topicโit's a fundamental tool that extends into university-level mathematics. Mastering this method will unlock a plethora of applications across various fields of study.
The Method Explained
In this course, you'll discover:
- How Completing the Square Works: Learn the step-by-step process that turns a quadratic equation into a perfect square, revealing its roots.
- Why It's Important: Understand the significance of completing the square and how it can be applied in advanced mathematical concepts.
With geometrical illustrations, you'll see the method come to life, offering a clear visual representation of the concept and why it's named as such.
Applications of Completing the Square
Get ready to explore the vast applications of completing the square in mathematics:
- Elementary Applications: Solve quadratic equations, plot second degree polynomials on a graph, and more.
- Advanced Applications: Identify quadratic curves and surfaces, determine the definiteness of square matrices (or corresponding quadratic forms), and optimization techniques for functions with multiple variables.
Though we won't cover these advanced applications in detail within this course due to time constraints, rest assuredโwe'll guide you to the courses where you can delve deeper into theory and practice. You'll also get hands-on practice with completing the square.
A Note on Advanced Calculus Applications
Did you know that completing the square is a crucial step in integrating rational functions during Calculus 2? After breaking down the function into simpler fractions, especially those with second-degree polynomials as denominators, you'll often need to complete the square to perform correct variable substitutions. For a thorough exploration of polynomial decomposition and integration using arctan, consider enrolling in my courses "Precalculus 2: Polynomials and Rational Functions" and "Calculus 2, part 1 of 2: Integrals with Applications."
Exploring Further
This course is just the beginning. You'll learn about all the courses we offer, which cover a wide range of topics in mathematics from high school to university level. We're also excited to share our plans for future courses, including approximate (and very hypothetical!) release dates for new content that will continue to advance your mathematical skills and knowledge.
Join us on this mathematical adventure where "Completing the Square" is not just a high school memory but a powerful skill for your university education and beyond! ๐๐
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Our review
๐ Overall Course Review ๐
The online course on 'Completing the Square' has received an overwhelmingly positive response from its participants, with a global rating of 4.95 out of 5 stars. The recent reviews highlight the course's effectiveness in solidifying skills, its comprehensive coverage of the topic, and the clarity and thoroughness of the explanations provided by the instructor. Here's a detailed breakdown of the feedback:
Course Content and Quality of Explanation:
Pros:
- ๐ซ Solid Foundational Knowledge: The course is designed to help students understand and apply the concept of completing the square, which some found was not adequately covered in their previous educational experiences.
- โจ Real-World Applications: The course includes a variety of applications and examples, making the topic relevant and practical for real-world problem-solving.
- ๐ง Clear Explanations: The instructor's explanations are highly praised for being clear, logical, and easy to follow, which is particularly beneficial for understanding new concepts without dwelling excessively on known ones.
- ๐ค Engaging Content: The course covers both two and three variable problems, offering a deeper dive into the subject matter that goes beyond what most users will ever need.
- ๐ Broad Contextual Relevance: The course doesn't just focus on algebra; it also includes contexts such as graphing conics, quadric surfaces, and quadratic forms, catering to students with varying levels of prior knowledge.
- ๐ช Challenging Problems: The problems and examples range from simple to complex, incorporating fractions and awkward coefficients, which is appreciated for its authenticity and challenge.
Cons:
- ๐ Advanced Topics: Some users wish that the course or similar content was available sooner, particularly for those who are looking for advanced topics like Discrete Math or Probability & Statistics.
Instructor's Performance:
Pros:
- ๐ Expertise and Attention to Detail: The instructor is noted for being highly knowledgeable, providing detailed explanations, and offering plenty of follow-through in their work.
- โค๏ธ Enthusiastic Feedback: One user expressed particularly strong praise for the instructor's enthusiasm and attention to detail, highlighting them as one of the best instructors they have had the pleasure of learning from.
Course Structure and Presentation:
Pros:
- ๐ Logical Flow: The course is structured in a way that ensures a logical progression through the material, making it easier to understand new concepts without feeling like previously learned concepts are being unnecessarily dragged out.
Course Impact and Relevance:
Pros:
- ๐ Skill Solidification: The course has helped many students solidify their skills in a way that they can now confidently share with others, such as colleagues at work.
- ๐ Long-Term Benefit: Users have noted that the comprehensive nature of the course will be beneficial throughout their academic and professional lives, even if some parts seemed beyond what they might immediately need.
Course Accessibility:
Pros:
- ๐ Free Access: The course is offered as a free resource, which is highly appreciated by the users and is seen as an excellent opportunity for those looking to flex their algebraic muscles or refresh their knowledge on the topic.
In summary, this 'Completing the Square' course is highly recommended for its comprehensive coverage of the subject, clear explanations, practical applications, and the instructor's expertise. It is a valuable resource that not only serves as a refresher but also provides deep insights into the applications of the concept, making it an essential tool for anyone looking to strengthen their algebraic foundation.