SOLUTION: Determine the vertex of the function by completing the square. -18x^2 + -177x + -190 An explanation of how to go about completing the square would be very much appreciate

Algebra ->  Algebra  -> Trigonometry-basics -> SOLUTION: Determine the vertex of the function by completing the square. -18x^2 + -177x + -190 An explanation of how to go about completing the square would be very much appreciate      Log On

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Question 215000: Determine the vertex of the function by completing the square.
-18x^2 + -177x + -190


An explanation of how to go about completing the square would be very much appreciated! Thanks!



Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!

-18x%5E2-177x-190 Start with the given expression.


-18%28x%5E2%2B%2859%2F6%29x%2B95%2F9%29 Factor out the x%5E2 coefficient -18. This step is very important: the x%5E2 coefficient must be equal to 1.


Take half of the x coefficient 59%2F6 to get 59%2F12. In other words, %281%2F2%29%2859%2F6%29=59%2F12.


Now square 59%2F12 to get 3481%2F144. In other words, %2859%2F12%29%5E2=%2859%2F12%29%2859%2F12%29=3481%2F144


-18%28x%5E2%2B%2859%2F6%29x%2Bhighlight%283481%2F144-3481%2F144%29%2B95%2F9%29 Now add and subtract 3481%2F144 inside the parenthesis. Make sure to place this after the "x" term. Notice how 3481%2F144-3481%2F144=0. So the expression is not changed.


-18%28%28x%5E2%2B%2859%2F6%29x%2B3481%2F144%29-3481%2F144%2B95%2F9%29 Group the first three terms.


-18%28%28x%2B59%2F12%29%5E2-3481%2F144%2B95%2F9%29 Factor x%5E2%2B%2859%2F6%29x%2B3481%2F144 to get %28x%2B59%2F12%29%5E2.


-18%28%28x%2B59%2F12%29%5E2-1961%2F144%29 Combine like terms.


-18%28x%2B59%2F12%29%5E2-18%28-1961%2F144%29 Distribute.


-18%28x%2B59%2F12%29%5E2%2B1961%2F8 Multiply.


So after completing the square, -18x%5E2-177x-190 transforms to -18%28x%2B59%2F12%29%5E2%2B1961%2F8. So -18x%5E2-177x-190=-18%28x%2B59%2F12%29%5E2%2B1961%2F8.


So -18x%5E2-177x-190=0 is equivalent to -18%28x%2B59%2F12%29%5E2%2B1961%2F8=0.


So y=-18x%5E2-177x-190 is equivalent to y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8.


So the equation y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 is now in vertex form y=a%28x-h%29%5E2%2Bk where a=-18, h=-59%2F12, and k=1961%2F8


Remember, the vertex of y=a%28x-h%29%5E2%2Bk is (h,k).


So the vertex of y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 is since h=-59%2F12 and k=1961%2F8