quadratics help?

Question

the equations f(x)=a(x-x1)(x-x2) and f(x)=a(x-x0)^2 +ho are related. How do i use algebra to relate the four parameters x1, x2, x0, ho. (suffices to express x0, h0 in terms of x1, and x2.)

Answer 1

f(x)=a(x-x1)(x-x2) and f(x)=a(x-x0)^2 +ho

the first one: ax^2 +ax(-x2-x1) +ax1x2
the second one: ax^2 - 2ax x0 + ax0^2 +h0

so:
2x0 = x1+x2
and
ax0^2 +h0 = ax1x2

substitute the first into the second:
a( (x1+x2)/2 )^2 +h0 = ax1`x2
a/4( x1^2 + 2x1x2+ x2^2 ) + h0 = a x1x2
a/4( x1^2 + 2x1x2+ x2^2 ) -ax1x2 = h0
~

Answer 2

Both equations are for the same parabola. For the left one the parabola crosses the x axis at x1 and x2. If you want the right one to cross at the same points, used quad eqn to get the roots, The two roots are where f(x) =0 . Alternately, piug x1 first and then x2 into the right equation, set it to 0 and solve for x0, h0.