Completing the Square
Theorem
Let $a, b, c, x$ be real numbers with $a \ne 0$.
Then:
- $a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$
This process is known as .
Proof
| \(\ds a x^2 + b x + c\) | \(=\) | \(\ds \frac {4 a^2 x^2 + 4 a b x + 4 a c} {4 a}\) | multiplying top and bottom by $4 a$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}\) |
$\blacksquare$
Also presented as
can also be presented in the form:
- $a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}$
Some present the result as a solution to the monic form $x^2 + b x + c$:
- $x^2 + b x + c = \paren {x + \dfrac b 2}^2 + c - \paren {\dfrac b 2}^2$
in this or a similar form.
Examples
Arbitrary Example
Consider the quadratic equation:
- $2 x^2 + 5 x + 1 = 0$
This is solved explicitly by as follows:
| \(\ds 2 x^2 + 5 x + 1\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x^2 + \dfrac 5 2\) | \(=\) | \(\ds -\dfrac 1 2\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x + \dfrac 5 4}^2\) | \(=\) | \(\ds -\dfrac 1 2 + \paren {\dfrac 5 4}^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac 1 2 + \dfrac {25} {16}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {17} {16}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + \dfrac 5 4\) | \(=\) | \(\ds \dfrac {\pm \sqrt {17} } 4\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {-5 \pm \sqrt {17} } 4\) |
Also see
- Definition:Tschirnhaus Transformation
- Quadratic Formula
Historical Note
The technique of was known to the ancient Babylonians as early as $1600$ BCE.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): complete the square
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): completing the square
- 2004: Ian Stewart: Galois Theory (3rd ed.) ... (previous) ... (next): Historical Introduction: Polynomial Equations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): completing the square
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): completing the square
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadratic equation
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): completing the square