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Mathematics: Completing the square

Name: Mathematics: Completing the square
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Rating: 4.9 (235 reviews)

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Korshub's take

Master one of algebra's most versatile techniques

Completing the square is a deceptively simple method that shows up again and again in more advanced mathematics. This course explains not just the mechanical steps but the reasoning behind them, using geometric illustrations to show why the method — and its name — make sense.

Who it's for

Students anywhere from high school to university who want a solid, visual understanding of the technique rather than a memorized formula.

Key takeaways

How and why the method of completing the square works, illustrated geometrically
Elementary applications: solving quadratic equations and sketching second-degree polynomial graphs
A guided tour of where the technique reappears later — quadratic curves and surfaces, definiteness of square matrices, and optimization in several variables
Practice exercises to build fluency

Korshub's take: the real value here is the map forward — knowing exactly which future courses lean on this one skill makes it easier to see why it's worth mastering now.

About this course

Mathematics: Completing the square

Mathematics from high school to university

1. Completing the square: how the method works, and why

You will learn about the method of completing the square, how it works and why. This remarkable (and elementary!) method has surprisingly many applications in advanced mathematics, and this is why it is really good to master it. Geometrical illustrations will give you a nice visual explanation of both the method and the name of the method.

2. A glimpse into some applications of completing the square

You will learn about various applications of completing the square, starting with quite elementary applications (high-school level) such as solving quadratic equations and drawing graphs of second degree polynomials, and continuing with some information about more advanced applications such as identifying quadratic curves and surfaces, determining definiteness of square matrices (or corresponding quadratic forms), and optimization of functions in two or more variables. These applications will not be treated in this course due to time constraints, but you will get information in which courses you find both theory and practice on the topics. You will also get some practice in completing the square. In this brief course I have chosen to inform you about applications of completing the square in Algebra, Calculus, and Linear Algebra and Geometry.

A note: Another typical application of completing the square (not discussed in this course) is in Calculus 2, when you must integrate a rational function (a function of the type p(x)/q(x), where both p(x) and q(x) are polynomials). After performing partial fraction decomposition, you will represent your function as a sum of simpler fractions, with denominators being first-degree polynomials or second-degree polynomials without real zeros. This second type, when integrated, will lead to some function defined by arctan; in order to perform this integration in a correct way (variable substitution), you must... complete the square! You can learn about partial fraction decomposition in my course "Precalculus 2: Polynomials and rational functions", and about the integration method mentioned above in my upcoming course "Calculus 2, part 1 of 2: Integrals with applications".