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Mathematics: Completing the square
Rating: 4.9 out of 5(236 ratings)
8,884 students

Mathematics: Completing the square

Mathematics from high school to university
Created byHania Uscka-Wehlou, Martin Wehlou
Last updated 5/2026
English
English [Auto],

What you'll 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.

Explore related topics

Course content

3 sections • 12 lectures • 1h 55m total length

  • Introduction to the course2:54

    Learn completing the square, why the method works, and its basic applications in a free two-hour course with an introduction to further precalculus topics.

  • Computation rules for the real numbers11:31

    Explore the computation rules for real numbers, including commutative, associative, and distributive laws, neutral and inverse elements, zero product property, and their use in solving polynomials and completing the square.

  • Some consequences of the distributive law6:33

    Demonstrate how to multiply two sums using the distributive and commutative laws to obtain the full expansion of (a+b)(c+d) for all real numbers a, b, c, d.

  • Three important formulas with geometrical illustrations12:47

    Explore three essential formulas with geometric illustrations: the square of a sum, the square of a difference, and the difference of two squares, with proofs and completing the square insights.

  • Completing the square in one variable15:02

    Learn to complete the square in one variable by turning a trinomial into a single squared term plus a constant, using the square of a sum/difference, with worked examples.

  • Completing the square in one variable at a time9:59

    Complete the square in multiple variables, one at a time, by separating x, y, and z. The process reveals a circle centered at (1,-2) and a cone.

  • Completing the square in two variables15:28

    Complete the square in two variables by rewriting a second-degree form as a sum of independent squares, using a symmetric matrix, and compare methods starting from x or from y.

  • Completing the square in three variables12:53

    Learn to complete the square for a three-variable quadratic form using a symmetric 3x3 matrix, and express it as a linear combination of squares in new variables, revealing positive definiteness.

Requirements

  • Being able to perform operations of addition and multiplication of real numbers.

Description

Mathematics: Completing the square

Mathematics from high school to university


[None of our courses are produced using AI; they are all real-human products.]


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".

Who this course is for:

  • High school students who want to learn the method of completing the square and who are curious about its applications in university topics.
  • University or college students who have discovered that they need the method of completing the square for some university level courses, and they want to re-learn this method.
  • Everybody who wants to learn a remarkable but elementary method, which has surprisingly many applications in various advanced branches of mathematics.

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