SOLUTION: how do i put -3x^2+6x+9 into a(x-h)^+k form?!?!

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Question 588865: how do i put -3x^2+6x+9 into a(x-h)^+k form?!?!
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-3x%5E2%2B6x%2B9 Start with the given expression.


-3%28x%5E2-2x-3%29 Factor out the x%5E2 coefficient -3. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient -2 to get -1. In other words, %281%2F2%29%28-2%29=-1.


Now square -1 to get 1. In other words, %28-1%29%5E2=%28-1%29%28-1%29=1


-3%28x%5E2-2x%2Bhighlight%281-1%29-3%29 Now add and subtract 1 inside the parenthesis. Make sure to place this after the "x" term. Notice how 1-1=0. So the expression is not changed.


-3%28%28x%5E2-2x%2B1%29-1-3%29 Group the first three terms.


-3%28%28x-1%29%5E2-1-3%29 Factor x%5E2-2x%2B1 to get %28x-1%29%5E2.


-3%28%28x-1%29%5E2-4%29 Combine like terms.


-3%28x-1%29%5E2-3%28-4%29 Distribute.


-3%28x-1%29%5E2%2B12 Multiply.


So after completing the square, -3x%5E2%2B6x%2B9 transforms to -3%28x-1%29%5E2%2B12. So -3x%5E2%2B6x%2B9=-3%28x-1%29%5E2%2B12.


So y=-3x%5E2%2B6x%2B9 is equivalent to y=-3%28x-1%29%5E2%2B12.


So the equation y=-3%28x-1%29%5E2%2B12 is now in vertex form y=a%28x-h%29%5E2%2Bk where a=-3, h=1, and k=12

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+-3x%5E2+%2B+6x+%2B+9+
+a%2A%28+x+-+h+%29%5E2+%2B+k+
+a%2A%28+x%5E2+-+2%2Ah%2Ax+%2B+h%5E2+%29+%2B+k+
+a%2Ax%5E2+-+2%2Aa%2Ah%2Ax+%2B+a%2Ah%5E2+%2B+k+
(1) +a+=+-3+
(2) +-2%2Aa%2Ah+=+6+
(3) +a%2Ah%5E2+%2B+k+=+9+
This is 3 equations and 3 unknowns, so
it should be solvable
Plug (1) into (2)
(2) +-2%2A%28-3%29%2Ah+=+6+
(2) +6h+=+6+
(2) +h+=+1+
and
(3) +-3%2A1%5E2+%2B+k+=+9+
(3) +-3+%2B+k+=+9+
(3) +k+=+12+
so,
+-3x%5E2+%2B+6x+%2B+9++=+-3%2A%28+x+-+1+%29%5E2+%2B+12+
check answer:
++-3%2A%28+x+-+1+%29%5E2+%2B+12+
+-3%2A%28+x%5E2+-+2x+%2B+1+%29+%2B+12+
+-3x%5E2+%2B+6x+-+3+%2B+12+
+-3x%5E2+%2B+6x+%2B+9+
OK