How do I find a quadratic equation given only both zeroes and the vertex's y coordinate?

I like the way geno did this one.......however..... what if we forgot (or didn't know) that the x coordinate of the vertex occurs mid-way between the roots?? Here's an alternative approach - though maybe not as "neat"

We have

0 = a(10)^2 + b(10) + c

0 = a(34)^2 + b(34) + c

Subtracting the first equation from the second, we get

0= a(34^2 - 10^2) + b(34 - 10)  simplify

0 = a(1056) + b(24)    rearrange

-b(24) = a(1056)       divide both sides by 24

-b = a(44) = 44a

Now.....suppose that we do remember that the x coordinate of the vertex is given by

x = -b / 2a    so we have

x = 44a / 2a  =  22   .....and that's the x coordinate of the vertex

And now we can continue the problem just as geno did by finding the value of "a," etc.  !!!

The x-value of the vertex will be half-ways between the two zeroes. Half-ways between x = 10 and x = 34         is x = 22; so the vertex is the point (22, 24).

An equation of a parabola is:  y - k  =  a(x - h)²  where the vertex is (h, k) = (22, 24).

--->   y - 24  =  a(x - 22)²       Since a point on the graph is (10, 0), replaceng x with 10 and y with 0:

--->   0 - 24  =  a(10 - 22)² 

--->   -24  =  a(-12)² 

--->   -24  =  144a

--->   a  =  =1/6

Equaiton is:  y - 24  =  (-1/6)(x - 22)²