Question 760085
I'm hoping this answers your question.
{{{ax^2+bx+c=0}}} is one of infinite equivalent equations, and you may not need to find {{{a}}}.
If you know that the solutions of a quadratic equation are {{{x=c}}} and {{{x=d}}}, you can conclude that one of the equivalent equations with those solutions is
{{{(x-c)(x-d)=0}}} which can also be written as {{{x^2-(c+d)x+cd=0}}}
If you multiply both sides of the equal sign times a non-zero number, you get an equivalent equation, such as
{{{2x^2-2(c+d)x+2cd=0}}} if you multiply times {{{2}}}, or
{{{ax^2-a(c+d)x+acd=0}}} if you multiply times a number {{{a}}} such that {{{a<>0}}}.
 
On the other hand, If you are looking for a quadratic function
{{{y=ax^2+bx+c}}}, you need at least 3 (x,y) points (or equivalent information) to find {{{a}}}, {{{b}}}, and {{{c}}}.
The coordinates of the vertex and one other point would do too.