SOLUTION: solve equation by completing the square n^2+16n-97=-3

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Question 201572: solve equation by completing the square
n^2+16n-97=-3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
n%5E2%2B16n-97=-3 Start with the given equation.


n%5E2%2B16n-97%2B3=0 Add 3 to both sides.


n%5E2%2B16n-94=0 Combine like terms.

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Now let's complete the square for the expression n%5E2%2B16n-94


n%5E2%2B16n-94 Start with the given expression.


Take half of the n coefficient 16 to get 8. In other words, %281%2F2%29%2816%29=8.


Now square 8 to get 64. In other words, %288%29%5E2=%288%29%288%29=64


n%5E2%2B16n%2Bhighlight%2864-64%29-94 Now add and subtract 64. Make sure to place this after the "n" term. Notice how 64-64=0. So the expression is not changed.


%28n%5E2%2B16n%2B64%29-64-94 Group the first three terms.


%28n%2B8%29%5E2-64-94 Factor n%5E2%2B16n%2B64 to get %28n%2B8%29%5E2.


%28n%2B8%29%5E2-158 Combine like terms.


So after completing the square, n%5E2%2B16n-94 transforms to %28n%2B8%29%5E2-158. So n%5E2%2B16n-94=%28n%2B8%29%5E2-158.


So n%5E2%2B16n-94=0 is equivalent to %28n%2B8%29%5E2-158=0.


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Now let's solve n%5E2%2B16n-94=0


%28n%2B8%29%5E2-158=0 Start with the given equation.


%28n%2B8%29%5E2=0%2B158 Add 158 to both sides.


%28n%2B8%29%5E2=158 Combine like terms.


x%2B8=%22%22%2B-sqrt%28158%29 Take the square root of both sides.


n%2B8=sqrt%28158%29 or n%2B8=-sqrt%28158%29 Break up the "plus/minus" to form two equations.


n=-8%2Bsqrt%28158%29 or n=-8-sqrt%28158%29 Subtract 8 from both sides.


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Answer:


So the solutions are n=-8%2Bsqrt%28158%29 or n=-8-sqrt%28158%29.