Completing the Square
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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 completing the square.
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
This result can also be presented in the form:
- $a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}$
Also see
Historical Note
The technique of Completing the Square 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