${ x }^{ 2 }\pm bx+ac=0$

Take the equation ${ ax }^{ 2 }+bx+c=0$ .

In order to factor this equation, you need to find two values $d$ and $e$ where $\frac { a }{ d } =\frac { e }{ c }$ and $d+e=b$ . This means that $d=b-e$ , which means that $\frac { a }{ b-e } =\frac { e }{ c }$ , or that ${ e }^{ 2 }-be+ac=0$ . Similarly, it also means that ${ d }^{ 2 }-bd+ac=0$ .

By applying this process to ${ x }^{ 2 }-bx+ac=0$ ( $f\left( a,b,c \right) ={ x }^{ 2 }-bx+ac$ ), you end up with ${ x }^{ 2 }+bx+ac=0$ , then back to ${ x }^{ 2 }-bx+ac=0$ . At this point, it just cycles between the two equations.

I haven't thought much about these two equations, but the fact that the equation ${ ax }^{ 2 }+bx+c=0$ feeds into this 2-cycle seems somewhat strange to me. Can anyone explain why this happens?

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

x2±bx+ac=0{ x }^{ 2 }\pm bx+ac=0x2±bx+ac=0

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${ x }^{ 2 }\pm bx+ac=0$