Why are equations written by equating something to zero?

At least for quadratics, if you want to solve (for example) $x^2 + 5x +8 = 2$, it is much easier to subtract 2 from each side, and factor:

$x^2 +5x +8-2 =0$

$x^2 + 5x +6 =0$

$(x+2)(x+3)=0$

Here is the key: the only way for a product of numbers ($x+2$) and ($x+3$) to be equal to zero is for one to be zero. This is a property unique to zero, and explains (at least in part) why we often set equations equal to zero.

Once upon a time, mathematicians studied three different kinds of quadratic equations:

• $ax^2 + bx = c$
• $ax^2 + c = bx$
• $ax^2 = bx + c$

(I'm not sure if they studied the fourth case, since the solutions would be negative numbers)

Correspondingly, you had to learn three different methods for solving a quadratic equation! Quite annoying! By normalizing the equation to just a single form,

$$ ax^2 + bx + c = 0$$

you only have to learn one one method to solve all quadratic equations! I think this choice of the four possibilities is the least ad-hoc choice: many different sorts equations forms share an "something equals zero" version, when they might not have anything else in common.

It's because of the constant term. Look at it this way:

If you have $ax+b=c$, then $ax+(b-c)=0$ and there's some $d=b-c$ so that $ax+d=0$. Likewise, when $ax^2+bx+c=d$, $ax^2+cx+e=0$, where $e=c-d$. It's just a standard way of writing equations so that they are easier to deal with, categorize, and solve.

It's simply a way of putting an equation into a standard form. You can always add and subtract the same quantities from both sides so that one of the sides becomes zero without changing the solution(s) of the equation.