Completing the square

Animation depicting the process of completing the square. (Details, animated GIF version)

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠ $\textstyle ax^{2}+bx+c$ ⁠ to the form ⁠ $\textstyle a(x-h)^{2}+k$ ⁠ for some values of ⁠ $h$ ⁠ and ⁠ $k$ ⁠.^[1] In terms of a new quantity ⁠ $x-h$ ⁠, this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠ $\textstyle (x-h)^{2}$ ⁠ 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 ⁠ $x$ ⁠ represents an unknown length. Then the quantity ⁠ $\textstyle x^{2}$ ⁠ represents the area of a square of side ⁠ $x$ ⁠ and the quantity ⁠ ${\tfrac {b}{a}}x$ ⁠ represents the area of a pair of congruent rectangles with sides ⁠ $x$ ⁠ and ⁠ ${\tfrac {b}{2a}}$ ⁠. To this square and pair of rectangles one more square is added, of side length ⁠ ${\tfrac {b}{2a}}$ ⁠. This crucial step completes a larger square of side length ⁠ $x+{\tfrac {b}{2a}}$ ⁠.

Completing the square is the oldest method of solving general quadratic equations, used in 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 functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent,^[2] and finding Laplace transforms.^[3]^[4]

^ Anita Wah; Creative Publications, Inc (1994). Algebra: Themes, Tools, Concepts. Henri Picciotto. p. 500. ISBN 978-1-56107-251-4. Extract of page 500
Chris Kornegay (1999). Math Dictionary With Solutions. SAGE. p. 373. ISBN 978-0-7619-1785-4. Extract of page 373
The form $a(x+h)^{2}+k$ is also sometimes used.
Karen Morrison; Nick Hamshaw (2018). Cambridge IGCSE® Mathematics Core and Extended Coursebook (illustrated, revised ed.). Cambridge University Press. p. 322. ISBN 978-1-108-43718-9. Extract of page 322
Shefiu Zakariyah (2024). Foundation Mathematics for Engineers and Scientists with Worked Examples. Taylor & Francis. p. 254. ISBN 978-1-003-85984-0. Extract of page 254
^ Dionissios T. Hristopulos (2020). Random Fields for Spatial Data Modeling: A Primer for Scientists and Engineers. Springer Nature. p. 267. ISBN 978-94-024-1918-4. Extract of page 267
^ James R. Brannan; William E. Boyce (2015). Differential Equations: An Introduction to Modern Methods and Applications (3rd ed.). John Wiley & Sons. p. 314. ISBN 978-1-118-98122-1. Extract of page 314
^ Stephen L. Campbell; Richard Haberman (2011). Introduction to Differential Equations with Dynamical Systems (illustrated ed.). Princeton University Press. p. 214. ISBN 978-1-4008-4132-5. Extract of page 214

[1] Anita Wah; Creative Publications, Inc (1994). Algebra: Themes, Tools, Concepts. Henri Picciotto. p. 500. ISBN 978-1-56107-251-4. Extract of page 500
Chris Kornegay (1999). Math Dictionary With Solutions. SAGE. p. 373. ISBN 978-0-7619-1785-4. Extract of page 373
The form $a(x+h)^{2}+k$ is also sometimes used.
Karen Morrison; Nick Hamshaw (2018). Cambridge IGCSE® Mathematics Core and Extended Coursebook (illustrated, revised ed.). Cambridge University Press. p. 322. ISBN 978-1-108-43718-9. Extract of page 322
Shefiu Zakariyah (2024). Foundation Mathematics for Engineers and Scientists with Worked Examples. Taylor & Francis. p. 254. ISBN 978-1-003-85984-0. Extract of page 254

[2] Dionissios T. Hristopulos (2020). Random Fields for Spatial Data Modeling: A Primer for Scientists and Engineers. Springer Nature. p. 267. ISBN 978-94-024-1918-4. Extract of page 267

[3] James R. Brannan; William E. Boyce (2015). Differential Equations: An Introduction to Modern Methods and Applications (3rd ed.). John Wiley & Sons. p. 314. ISBN 978-1-118-98122-1. Extract of page 314

[4] Stephen L. Campbell; Richard Haberman (2011). Introduction to Differential Equations with Dynamical Systems (illustrated ed.). Princeton University Press. p. 214. ISBN 978-1-4008-4132-5. Extract of page 214

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