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 $ax^{2}+bx+c$ to the form $a(x-h)^{2}+k$ for some values of h and k.

In other words, completing the square places a perfect square trinomial inside of a quadratic expression.

Completing the square is used in

solving quadratic equations,
deriving the quadratic formula,
graphing quadratic functions,
evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent,^[1]
finding Laplace transforms.^[2]^[3]

In mathematics, completing the square is often applied in any computation involving quadratic polynomials.

^ 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] 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

[2] 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

[3] 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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