Editing Completing the square (section)

{{short description|Method for solving quadratic equations}}
[[File:Completing the square.ogv|thumb|Animation depicting the process of completing the square. ([[:File:Completing the square.ogv|Details]], [[:File:Completing the square.gif|animated GIF version]])]]

In [[elementary algebra]], '''completing the square''' is a technique for converting a [[quadratic polynomial]] of the form {{tmath|\textstyle ax^2 + bx + c}} to the form {{tmath|\textstyle a(x-h)^2 + k}} for some values of {{tmath|h}} and {{tmath|k}}.<ref>{{cite book |title=Algebra: Themes, Tools, Concepts |author1=Anita Wah |author2=Creative Publications, Inc |edition= |publisher=Henri Picciotto |year=1994 |isbn=978-1-56107-251-4 |page=500 |url=https://books.google.com/books?id=nejW803k3fYC}} [https://books.google.com/books?id=nejW803k3fYC&pg=PA500 Extract of page 500] {{pb}}
{{cite book |title=Math Dictionary With Solutions |author1=Chris Kornegay |edition= |publisher=SAGE |year=1999 |isbn=978-0-7619-1785-4 |page=373 |url=https://books.google.com/books?id=Mjo5DQAAQBAJ}} [https://books.google.com/books?id=Mjo5DQAAQBAJ&pg=PA373 Extract of page 373] {{pb}}
The form <math>a(x+h)^2 + k</math> is also sometimes used. {{pb}} {{cite book |title=Cambridge IGCSE® Mathematics Core and Extended Coursebook |author1=Karen Morrison |author2=Nick Hamshaw |edition=illustrated, revised |publisher=Cambridge University Press |year=2018 |isbn=978-1-108-43718-9 |page=322 |url=https://books.google.com/books?id=6M1MDwAAQBAJ}} [https://books.google.com/books?id=6M1MDwAAQBAJ&pg=PA322 Extract of page 322] {{pb}} {{cite book |title=Foundation Mathematics for Engineers and Scientists with Worked Examples |author1=Shefiu Zakariyah |edition= |publisher=Taylor & Francis |year=2024 |isbn=978-1-003-85984-0 |page=254 |url=https://books.google.com/books?id=8IooEQAAQBAJ}} [https://books.google.com/books?id=8IooEQAAQBAJ&pg=PA254 Extract of page 254]</ref> In terms of a new quantity {{tmath|x-h}}, this expression is a quadratic polynomial with no linear term. By subsequently isolating {{tmath|\textstyle (x-h)^2}} and taking the [[square root]], a quadratic problem can be reduced to a linear problem.

The name ''completing the square'' comes from a geometrical picture in which {{tmath|x}} represents an unknown length. Then the quantity {{tmath|\textstyle x^2}} represents the area of a [[square]] of side {{tmath|x}} and the quantity {{tmath|\tfrac{b}{a}x}} represents the area of a pair of [[Congruence (geometry)|congruent]] [[rectangle]]s with sides {{tmath|x}} and {{tmath|\tfrac{b}{2a} }}. To this square and pair of rectangles one more square is added, of side length {{tmath|\tfrac{b}{2a} }}. This crucial step ''completes'' a larger square of side length {{tmath|x + \tfrac{b}{2a} }}.

Completing the square is the oldest method of solving general [[quadratic equation]]s, used in [[Old Babylonian Empire|Old Babylonian]] clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra courses today. It is also used for graphing [[quadratic function]]s, deriving the [[quadratic formula]], and more generally in computations involving quadratic polynomials, for example in [[calculus]] evaluating [[Gaussian integral]]s with a linear term in the exponent,<ref>{{cite book |title=Random Fields for Spatial Data Modeling: A Primer for Scientists and Engineers |author1=Dionissios T. Hristopulos |edition= |publisher=Springer Nature |year=2020 |isbn=978-94-024-1918-4 |page=267 |url=https://books.google.com/books?id=wivRDwAAQBAJ}} [https://books.google.com/books?id=wivRDwAAQBAJ&pg=PA267 Extract of page 267]</ref> and finding [[Laplace transform]]s.<ref>{{cite book |title=Differential Equations: An Introduction to Modern Methods and Applications |author1=James R. Brannan |author2=William E. Boyce |edition=3rd |publisher=John Wiley & Sons |year=2015 |isbn=978-1-118-98122-1 |page=314 |url=https://books.google.com/books?id=Sy2oDwAAQBAJ}} [https://books.google.com/books?id=Sy2oDwAAQBAJ&pg=PA314 Extract of page 314]</ref><ref>{{cite book |title=Introduction to Differential Equations with Dynamical Systems |author1=Stephen L. Campbell |author2=Richard Haberman |edition=illustrated |publisher=Princeton University Press |year=2011 |isbn=978-1-4008-4132-5 |page=214 |url=https://books.google.com/books?id=Mt3nI-lQKZQC}} [https://books.google.com/books?id=Mt3nI-lQKZQC&pg=PA214 Extract of page 214]</ref>