Question about factoring/condensing equation rules

Question

I have the equation $x^2 - 6x = 72$ and then $x^2 - 6x - 72 = 0$ that's supposed to turn into $(x-12)(x+6)$. 72/6 = 12. So could just do that with any equation that? Divide the end thing with the middle coefficient of x to get the thing for the first set of parentheses? Or is it just a coincidence?

Answer 1

$ x^2+4x+4=0$

$(x+2)\cdot (x+2)=0$

$4÷4\neq 2$

It was coincidence.

Answer 2

This is a coincidence.

When the solutions are $a,b$:

• the middle term is $-a-b$ (here it is $-(-6)-12 = -6)$)
• the end term is $ab$ (here: $-6\times 12 = -72$).

The formula $$ -\frac{a+b}{ab} $$is not a solution in general.

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This is a coincidence because here you have $-a-b = a$.