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So the quadratic formula and completing the square give the same answer so

$ax^2 + bx + c \Rightarrow \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}$ Right?

So making a formula for completing the square should lead me to the quadratic formula

1. $$ax^ + bx + c \Rightarrow x^2 + \frac {bx + c}{a}$$
2. $$x^2 = \frac {bx}{a} + \frac {c}{a} \Rightarrow (x + \frac {b}{2a})^2 - \frac {b^2}{4a^2} + \frac {c}{a}$$
3. $$(x + \frac {b}{2a})^2 - \frac {b^2}{4a^2} + \frac {c}{a} \Rightarrow (x + \frac {b}{2a})^2 - \frac {b^2 + 4ac}{4a^2}$$

If we assume that $$ax^2 + bx + c = 0$$ Then

1. $$(x + \frac {b}{2a})^2 = \frac {b^2 + 4ac}{4a^2}$$
2. $$x + \frac {b}{2a} = \pm \sqrt {\frac {b^2 + 4ac}{4a^2}} \Rightarrow \frac {\pm \sqrt {b^2 + 4ac}}{\pm \sqrt {4a^2}}$$
3. $$x + \frac {b}{2a} = \frac {\pm \sqrt {b^2 - 4ac}}{2a}$$

So this then leads to

$x = \frac {-b \pm \sqrt {b^2 -4ac}}{2a}$

Note by Jack Rawlin
2 years, 3 months ago