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1) Derivation of the quadratic formula

This is note \(1\) in a set of notes showing how to obtain formulas. There will be no words beyond these short paragraphs as the rest will either consist of images or algebra showing the steps needed to derive the formula mentioned in the title.

Suggestions for other formulas to derive are welcome, however whether they are completed or not depends on my ability to derive them. The suggestions given aren't guaranteed to be the next one in the set but they will be done eventually.


1 \[\large ax^2 + bx + c = 0\]


2 \[\large x^2 + \frac{b}{a}x + \frac{c}{a} = 0\]


3.1 \[\large x^2 + \frac{b}{a}x = \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\]


3.2 \[\large \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0\]


4 \[\large \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} + \frac{4ac}{4a^2} = 0\]


5 \[\large \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}\]


6 \[\large x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}}\]


7 \[\large x + \frac{b}{2a} = \frac{\pm\sqrt{b^2 - 4ac}}{2a}\]


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

Note by Jack Rawlin
8 months, 4 weeks ago

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Yup! This is the derivation of the quadratic formula. Great! Pi Han Goh · 8 months, 4 weeks ago

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