SOLUTION: Solve this equation by completing the square: -4x^2 + 8x=7 Please help me with this.

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Question 78248: Solve this equation by completing the square:
-4x^2 + 8x=7
Please help me with this.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-4x%5E2+%2B+8x=7
-4x%5E2+%2B+8x-7=0 Subtract 7 from both sides
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-4+x%5E2%2B8+x-7 Start with the given equation



y%2B7=-4+x%5E2%2B8+x Add 7 to both sides



y%2B7=-4%28x%5E2-2x%29 Factor out the leading coefficient -4



Take half of the x coefficient -2 to get -1 (ie %281%2F2%29%28-2%29=-1).


Now square -1 to get 1 (ie %28-1%29%5E2=%28-1%29%28-1%29=1)





y%2B7=-4%28x%5E2-2x%2B1-1%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1 does not change the equation




y%2B7=-4%28%28x-1%29%5E2-1%29 Now factor x%5E2-2x%2B1 to get %28x-1%29%5E2



y%2B7=-4%28x-1%29%5E2%2B4%281%29 Distribute



y%2B7=-4%28x-1%29%5E2%2B4 Multiply



y=-4%28x-1%29%5E2%2B4-7 Now add %2B7 to both sides to isolate y



y=-4%28x-1%29%5E2-3 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-4, h=1, and k=-3. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-4x%5E2%2B8x-7 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-4x%5E2%2B8x-7%29 Graph of y=-4x%5E2%2B8x-7. Notice how the vertex is (1,-3).



Notice if we graph the final equation y=-4%28x-1%29%5E2-3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-4%28x-1%29%5E2-3%29 Graph of y=-4%28x-1%29%5E2-3. Notice how the vertex is also (1,-3).



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





So now we have
-4%28x-1%29%5E2-3=0
-4%28x-1%29%5E2=3 Add 3 to both sides
%28x-1%29%5E2=3%2F-4 Divide both sides by -4
x-1=0%2B-sqrt%28-3%2F4%29 Take the square root of both sides
x=0%2B-sqrt%28-3%2F4%29%2B1 Add 1 to both sides
Since we cannot take the square root of a negative number, we must factor out a negative 1
x=0%2B-sqrt%28-1%29sqrt%283%2F4%29%2B1
Now we'll let i=sqrt%28-1%29 and say
x=0%2B-sqrt%283%2F4%29i%2B1
x=0%2B-sqrt%283%29%2F2%2Ai%2B1 Reduce the denominator
So our answer is
x=sqrt%283%29%2F2%2Ai%2B1 or x=-sqrt%283%29%2F2%2Ai%2B1