SOLUTION: Complete the square: 2x^2+10x+11=0

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Question 83343: Complete the square: 2x^2+10x+11=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=2+x%5E2%2B10+x%2B11 Start with the given equation



y-11=2+x%5E2%2B10+x Subtract 11 from both sides



y-11=2%28x%5E2%2B5x%29 Factor out the leading coefficient 2



Take half of the x coefficient 5 to get 5%2F2 (ie %281%2F2%29%285%29=5%2F2).


Now square 5%2F2 to get 25%2F4 (ie %285%2F2%29%5E2=%285%2F2%29%285%2F2%29=25%2F4)





y-11=2%28x%5E2%2B5x%2B25%2F4-25%2F4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25%2F4 does not change the equation




y-11=2%28%28x%2B5%2F2%29%5E2-25%2F4%29 Now factor x%5E2%2B5x%2B25%2F4 to get %28x%2B5%2F2%29%5E2



y-11=2%28x%2B5%2F2%29%5E2-2%2825%2F4%29 Distribute



y-11=2%28x%2B5%2F2%29%5E2-25%2F2 Multiply



y=2%28x%2B5%2F2%29%5E2-25%2F2%2B11 Now add 11 to both sides to isolate y



y=2%28x%2B5%2F2%29%5E2-3%2F2 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=-5%2F2, and k=-3%2F2. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=2x%5E2%2B10x%2B11 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2%2B10x%2B11%29 Graph of y=2x%5E2%2B10x%2B11. Notice how the vertex is (-5%2F2,-3%2F2).



Notice if we graph the final equation y=2%28x%2B5%2F2%29%5E2-3%2F2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x%2B5%2F2%29%5E2-3%2F2%29 Graph of y=2%28x%2B5%2F2%29%5E2-3%2F2. Notice how the vertex is also (-5%2F2,-3%2F2).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.