Zero to equations and inequalities

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    Inequalities Zero
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Discussion Overview

The discussion revolves around the reasoning behind setting equations and inequalities, particularly quadratic ones, to zero. Participants explore the historical and conceptual foundations of this practice, focusing on its utility in solving such equations through methods like factorization.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the rationale behind rewriting equations like x^2=5x+6 as x^2-5x-6=0, suggesting that it is primarily a matter of familiarity.
  • Another participant emphasizes the usefulness of having a right-hand side (RHS) of zero when solving equations in factorized form, hinting at its practical implications.
  • A participant expresses appreciation for the concept of factorizing quadratics set to zero, noting that it leverages the "zero property" of multiplication, which states that if the product of two numbers is zero, at least one of them must also be zero.
  • There is a reiteration of the "zero property" of multiplication, highlighting its significance in understanding the solutions to equations.

Areas of Agreement / Disagreement

Participants generally agree on the utility of setting equations to zero for solving them, particularly through factorization. However, the discussion includes varying perspectives on the underlying reasons and intuitiveness of this approach, indicating that multiple views remain.

Contextual Notes

Some assumptions regarding the familiarity with mathematical concepts and methods, such as factorization, are present but not explicitly stated. The discussion does not resolve the deeper historical context of why this practice was established.

Who May Find This Useful

Readers interested in the foundations of algebra, particularly in the context of solving quadratic equations and the reasoning behind mathematical conventions.

elmestlol
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Hello my question is why do we set some equations and inequalities to 0, for example quadratic equations/inequalities. I know that they should be 0, but why. How did people come with this when they invented it.
 
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Are you asking why
x^2=5x+6

will often be rewritten as
x^2-5x-6 = 0
?

The only reason is that it is familiar. There is no difference in the information content of these equivalent equations.
 
elmestlol, are you familiar with the method of solving simple quadratic equations using factorization?

Think carefully about why an RHS of zero is particularly useful for solving an equation in factorized form.
 
elmestlol, are you familiar with the method of solving simple quadratic equations using factorization?

Think carefully about why an RHS of zero is particularly useful for solving an equation in factorized form.
 
Yes I know it . Ok maybe I understand now why is it used.
 
I actually find the idea of factorizing quadratics that are set to zero very smart indeed. You can explain to anyone that if you multiply two numbers together and they equal zero, then either one or the other or both have to be zero. They reply with "well duh, obviously" but all of a sudden it becomes much less intuitive at first glance when solving quadratics.
 
it is the "zero property" of multiplication: If AB= 0 then either A= 0 or B= 0 or both.

But if AB= C where C is not 0, we know nothing about A and B separately.
 

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