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Completing the square formula

Going through completing the square with algebra gives the quadratic formula if you're not careful. So since completing the square gives an answer in the form of \(\pm \sqrt {y} - z\), we can work off of that to guide us.

  1. \(ax^2 + bx + c = 0 \Rightarrow x^2 + \frac {bx}{a} + \frac {c}{a} = 0\)

  2. \((x + \frac {b}{2a})^2 - \frac {b^2}{4a^2} + \frac {c}{a} = 0\)

  3. \((x + \frac {b}{2a})^2 = \frac {b^2}{4a^2} - \frac {c}{a}\)

  4. \(x + \frac {b}{2a} = \pm \sqrt {\frac {b^2}{4a^2} - \frac {c}{a}}\)

So that means that the formula for completing the square is

\[x = \pm \sqrt {\frac {b^2}{4a^2} - \frac {c}{a}} - \frac {b}{2a}\]

Note by Jack Rawlin
2 years, 10 months ago

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Well,actually this \(\textbf{IS}\) the quadratic formula.Just simplify: \[x=\pm\sqrt{\frac{b^2-4ac}{4a^2}}-\frac{b}{2a}\\x=\pm\frac{\sqrt{b^2-4ac}}{2a}-\frac{b}{2a}\\ \boxed{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}\]

Abdur Rehman Zahid - 2 years, 8 months ago

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