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Given a general polynomial of the form ax^2 + bx + c = 0, we can factor it to be the product of two terms eg. (x+d)*(x-e) = 0.

Why is it that we can use substitute BOTH equations as equalling zero? If one is zero then the other isn't necessarily.

Also, in many cases with any variables equaling zero, I have always understood that dividing by that variable would make it invalid or almost lose its integrity in a way.

For example if we did (x+d)(x-e) = 0 ---> (x+d)(x-e)/(x+d) = 0/(x+d). That does not necessarily equal zero. Is this only because x+d can make that term undefined at x = -d or is there a deeper meaning behind that variable being invalid?

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    $\begingroup$ I’m voting to close this question because it is better posted on the Mathematics stack. $\endgroup$
    – Solar Mike
    Commented Sep 11, 2023 at 5:54

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All this equation is asking is to define conditions under which our function is zero. X is an independent variable therefore any value of x that satisfies the equation means that it is a root. Generally, our logic fails because if one condition is met, it is only met when x equals that one value. We need to determine every value of x for which the equation is true to get all possible roots.

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