SOLUTION: x^2+6x=7 Solve by completing the square to obtain exact solutions.

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Question 77506: x^2+6x=7
Solve by completing the square to obtain exact solutions.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B6x=7

x%5E2%2B6x-7=0 Subtract 7 from both sides

y=x%5E2%2B6x-7 Set the expression equal to y

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=1+x%5E2%2B6+x-7 Start with the given equation



y%2B7=1+x%5E2%2B6+x Add 7 to both sides



y%2B7=1%28x%5E2%2B6x%29 Factor out the leading coefficient 1



Take half of the x coefficient 6 to get 3 (ie %281%2F2%29%286%29=3).


Now square 3 to get 9 (ie %283%29%5E2=%283%29%283%29=9)





y%2B7=1%28x%5E2%2B6x%2B9-9%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 9 does not change the equation




y%2B7=1%28%28x%2B3%29%5E2-9%29 Now factor x%5E2%2B6x%2B9 to get %28x%2B3%29%5E2



y%2B7=1%28x%2B3%29%5E2-1%289%29 Distribute



y%2B7=1%28x%2B3%29%5E2-9 Multiply



y=1%28x%2B3%29%5E2-9-7 Now add %2B7 to both sides to isolate y



y=1%28x%2B3%29%5E2-16 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=-3, and k=-16. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=1x%5E2%2B6x-7 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2%2B6x-7%29 Graph of y=1x%5E2%2B6x-7. Notice how the vertex is (-3,-16).



Notice if we graph the final equation y=1%28x%2B3%29%5E2-16 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x%2B3%29%5E2-16%29 Graph of y=1%28x%2B3%29%5E2-16. Notice how the vertex is also (-3,-16).



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






So our equation can be converted into vertex form:

y=%28x%2B3%29%5E2-16

0=%28x%2B3%29%5E2-16 Let y=0

16=%28x%2B3%29%5E2 Add 16 to both sides

sqrt%2816%29=sqrt%28%28x%2B3%29%5E2%29 Take the square root of both sides



Which means:

x%2B3=4%29 or x%2B3=-4%29

x%2B3-3=4-3%29 or x%2B3-3=-4-3%29 Subtract 3 from both sides for each case

So our solutions are:

x=1%29 or x=-7%29