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find the x-intercept of the parabola with vertex (6,27) and y-intercept (0,-81)

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The vertex form of the equation of a parabola with the vertex (h,k):
[tex]y=a(x-h)^2+k[/tex]

The parabola has the vertex at (6,27).
[tex]y=a(x-6)^2+27[/tex]

The y-intercept is (0,-81).
[tex]x=0, \ y=-81 \\ \Downarrow \\ -81=a(0-6)^2+27 \\ -81=a(-6)^2+27 \\ -81=36a+27 \\ -81-27=36a \\ -108=36a \\ \frac{-108}{36}=a \\ a=-3[/tex]

The equation of the parabola is:
[tex]y=-3(x-6)^2+27[/tex]

y=0 at the x-intercepts.
[tex]0=-3(x-6)^2+27 \\ -27=-3(x-6)^2 \\ \frac{-27}{-3}=(x-6)^2 \\ 9=(x-6)^2 \\ \sqrt{9}=\sqrt{(x-6)^2} \\ 3=|x-6| \\ x-6=3 \ \lor \ x-6=-3 \\ x=3+6 \ \lor \ x=-3+6 \\ x=9 \ \lor \ x=3[/tex]

The x-intercepts are x=3 and x=9.

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