how to you switch between these to forms completing the square?
to go from vertex form f(x) = a(x - h)^2 + k to standard ax^2 + bx + c form, just expand the squared term, distribute a, and combine like terms: ah^2 + k = c
to go from standard form to vertex form, you need to complete the square:
ax^2 + bx + c =
a(x^2 + b/a x ) + c =
a(x^2 + b/a x + (b/2a)^2 ) + c - a(b / 2a)^2 =
a(x + b / 2a)^2 + [c - a(b / 2a)^2] =
a(x + b / 2a)^2 + [(4ac - b) / (4a)]
so h = -b / 2a
and k = (4ac - b) / 4a = c - b / 4a
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Verified answer
to go from vertex form f(x) = a(x - h)^2 + k to standard ax^2 + bx + c form, just expand the squared term, distribute a, and combine like terms: ah^2 + k = c
to go from standard form to vertex form, you need to complete the square:
ax^2 + bx + c =
a(x^2 + b/a x ) + c =
a(x^2 + b/a x + (b/2a)^2 ) + c - a(b / 2a)^2 =
a(x + b / 2a)^2 + [c - a(b / 2a)^2] =
a(x + b / 2a)^2 + [(4ac - b) / (4a)]
so h = -b / 2a
and k = (4ac - b) / 4a = c - b / 4a