SOLUTION: Use the Completing the Square method to find the vertex form of the quadratic function y=2x^2+8x+18

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Question 192112: Use the Completing the Square method to find the vertex form of the quadratic function y=2x^2+8x+18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2x%5E2%2B8x%2B18 Start with the right side of the given equation.


2%28x%5E2%2B4x%2B9%29 Factor out the x%5E2 coefficient 2. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient 4 to get 2. In other words, %281%2F2%29%284%29=2.


Now square 2 to get 4. In other words, %282%29%5E2=%282%29%282%29=4


2%28x%5E2%2B4x%2Bhighlight%284-4%29%2B9%29 Now add and subtract 4 inside the parenthesis. Make sure to place this after the "x" term. Notice how 4-4=0. So the expression is not changed.


2%28%28x%5E2%2B4x%2B4%29-4%2B9%29 Group the first three terms.


2%28%28x%2B2%29%5E2-4%2B9%29 Factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2.


2%28%28x%2B2%29%5E2%2B5%29 Combine like terms.


2%28x%2B2%29%5E2%2B2%285%29 Distribute.


2%28x%2B2%29%5E2%2B10 Multiply.


So after completing the square, 2x%5E2%2B8x%2B18 transforms to 2%28x%2B2%29%5E2%2B10. So 2x%5E2%2B8x%2B18=2%28x%2B2%29%5E2%2B10.


So y=2x%5E2%2B8x%2B18 is equivalent to y=2%28x%2B2%29%5E2%2B10.


Now the equation y=2%28x%2B2%29%5E2%2B10 is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=-2, and k=10


Remember, the vertex of y=a%28x-h%29%5E2%2Bk is (h,k). So the vertex of y=2%28x%2B2%29%5E2%2B10 ( and 2x%5E2%2B8x%2B18) is (-2,10)