Hi, how do you complete this square x2+y2=2x
To get this result (x-1)2+y2=1
Write it as ##(x^2 - 2x\quad\quad)+y^2 = 0## Then figure out what number you can add to both sides to fill in the blank and make the quantity in parentheses a perfect square.
Using number properties to have each variable on either one side or the other, you obtain:
The expression on the left side is not a square; you WANT a square. The expression can be factored,
x(x-2) = y2.
Notice you can represent the left hand side as a rectangular area with a picture (if you wanted). One side is x long and the other side is x-2 long. Nevermind the negative or subtraction; but if you could split the "-2" in half and reposition one of those halves along the other part of the length, x, (this really needs a picture for you to see), then you would see a missing square piece on a corner. The area of this missing square piece is (2/2) by (2/2), or 1 by 1 square units. This is 1 square unit, to "complete the square".
Continuing then, add 1 to both sides of the equation:
As I said, a picture will help to show this completion of the square. Can you finish the problem from here?
EDIT: I posted this too quickly and I believe I made a sign mistake.
Ahh Yess If you add one one both sides it works :)
That's it. When you find the quantity, be sure to add it to both sides of the equation.
EDIT: Good, you found it already.
By the way, i always feel like i get competent understandable explanations in this forum. Are some of you teachers? Or just very devoted in helping others understand?
Although this already has been answered, I would just like to give something I believe is missing:
As said: x^2-2x = -y^2.
As said also, x^2-2x = x(x-2), now were looking for a perfect square, you can do any number, but easy ones will be (x-2)^2. Now we calculate (x-2)^2 = x^2-2x+4. So we need to add 4 to both sides, that is x^2-2x+4 = 4-y^2. Taking the square root (this is what we planned everything for, we made sure the left side will be a nice root):
(x-2)^2 = (2+y)(2-y), taking the square root yields:
x-2 = sqrt((2+y)(2-y)
so x = 2+ sqrt((2+y)(2-y)
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These steps aren't needed because the problem was to complete the square, not to solve for x.
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