SOLUTION: complete the square, and find the roots of the quadratic equation. x^2 - 2/3x = 0

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Question 48580: complete the square, and find the roots of the quadratic equation.
x^2 - 2/3x = 0
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
+%28x-a%29%5E2+ when multiplied out becomes +x%5E2+-+2ax+%2B+a%5E2+. Our aim is to find the value of a.

Looking at +x%5E2+-+%282%2F3%29x+ we have 2a=2/3. Hence a = 1/3.

So to get into the correct form we need to add on "a%5E2". This is 1/9. We therefore get +x%5E2+-+%282%2F3%29x+%2B+1%2F9+ but we can't just add 1/9 to the equation as that changes what the equation was, so we also add -1/9, so that 1/9-1/9 is zero and in total we have added "nothing":

+x%5E2+-+%282%2F3%29x+%2B+1%2F9+-+1%2F9+
+%28x%5E2+-+%282%2F3%29x+%2B+1%2F9%29+-+1%2F9+
+%28x+-+1%2F3%29%5E2+-+1%2F9+

And going back to the question:
+%28x+-+1%2F3%29%5E2+-+1%2F9+=+0+
+%28x+-+1%2F3%29%5E2+=+1%2F9+
+x+-+1%2F3+=+sqrt%281%2F9%29+ or +x+-+1%2F3+=+-sqrt%281%2F9%29+
+x+-+1%2F3+=+1%2F3+ or +x+-+1%2F3+=+-1%2F3+
+x+=+2%2F3+ or +x+=+0+
+x+=+2%2F3+ or +x+=+0+

Just so you are aware... for this question, completing the square is completely overkill. Much better to do the following:
+x%5E2+-+%282%2F3%29x+=+0+
x(x-(2/3)) = 0
so x=0 or x-(2/3)=0
--> x=0 or x=2/3

jon