SOLUTION: solve the equation by completing the square: 8x^2+4x-4=0

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Question 720791: solve the equation by completing the square:
8x^2+4x-4=0
Answer by MathLover1(20850) 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=8+x%5E2%2B4+x-4 Start with the given equation



y%2B4=8+x%5E2%2B4+x Add 4 to both sides



y%2B4=8%28x%5E2%2B%281%2F2%29x%29 Factor out the leading coefficient 8



Take half of the x coefficient 1%2F2 to get 1%2F4 (ie %281%2F2%29%281%2F2%29=1%2F4).


Now square 1%2F4 to get 1%2F16 (ie %281%2F4%29%5E2=%281%2F4%29%281%2F4%29=1%2F16)





y%2B4=8%28x%5E2%2B%281%2F2%29x%2B1%2F16-1%2F16%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1%2F16 does not change the equation




y%2B4=8%28%28x%2B1%2F4%29%5E2-1%2F16%29 Now factor x%5E2%2B%281%2F2%29x%2B1%2F16 to get %28x%2B1%2F4%29%5E2



y%2B4=8%28x%2B1%2F4%29%5E2-8%281%2F16%29 Distribute



y%2B4=8%28x%2B1%2F4%29%5E2-1%2F2 Multiply



y=8%28x%2B1%2F4%29%5E2-1%2F2-4 Now add %2B4 to both sides to isolate y



y=8%28x%2B1%2F4%29%5E2-9%2F2 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=8, h=-1%2F4, and k=-9%2F2. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=8x%5E2%2B4x-4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C8x%5E2%2B4x-4%29 Graph of y=8x%5E2%2B4x-4. Notice how the vertex is (-1%2F4,-9%2F2).



Notice if we graph the final equation y=8%28x%2B1%2F4%29%5E2-9%2F2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C8%28x%2B1%2F4%29%5E2-9%2F2%29 Graph of y=8%28x%2B1%2F4%29%5E2-9%2F2. Notice how the vertex is also (-1%2F4,-9%2F2).



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