How do you write a quadratic equation in standard form for a given set of zeros?

Answer 1

The standard form is
ax^2 + bc + c = 0

The value of ax^2 + bx +c will be zero when
x = [-b +/- sqrt (b^2-4ac)]/2a

If you want to write a quadratic equation that is zero at given values x=d and x=e, first write it as
(x-d)(x-e) = 0
and then multiply it out for the standard form:
x^2 -(d+e)x + de = 0

Answer 2

say you had (4,0) and (-3,0) given as zeros of a quadratic function.
Then x-4 = 0
plus x+3 = 0
and
(x-4)(x+3) = 0
and
x^2 - x - 12 = 0

I suppose I could generalize that to zeros at a and b
x-a = 0
x-b = 0
so
(x-a)(x-b) = 0
x^2 -(a+b)x + a b = 0