# given $x^2 + y^2 = 2x$. I want $(x-1)^2 + y^2 = 1$

given $$x^2 + y^2 = 2x$$. I want $$(x-1)^2 + y^2 = 1$$

Is this completing the square?

my attempt:

$$x^2 + y^2 - 2x = 0$$

need $$1$$

$$x^2 + y^2 - 2x + 1 = 1$$

I forgot how to could someone explain please

• $x^2+ax=(x+a/2)^2-a^2/4$ – kingW3 Apr 18 at 23:03
• Do you remember how to expand $(x-1)^2+y^2$? – Minus One-Twelfth Apr 18 at 23:03
• Yes. But how do you come up with it? – shah Apr 18 at 23:05
• expanding gives $x^2-2x+1+y^2=1$, which is the same as $x^2-2x+y^2$ that is $2x-2x=0$. so just go backwards to prove it – thesmallprint Apr 18 at 23:07
• Come up with what? The formula for $(a-b)^2$? To do this, just write it as $(a-b)(a-b)$ and use various laws (e.g. distributive and commutative), so it is $(a-b)a - (a-b)b = a^2 - ba - ab + b^2 = a^2-2ab+ b^2$. – Minus One-Twelfth Apr 18 at 23:09

This is indeed completing the square. In this case, you have $$x^2+y^2=2x$$ Since we want to complete the square of the variable $$x$$, we will group the $$x$$ terms. So we get $$x^2-2x+y^2=0$$ Recall that $$(a+b)^2=a^2+2ab+b^2$$ and $$(a-b)^2=a^2-2ab+b^2$$. We want to complete the square using the $$x$$ terms we already have, namely $$x^2-2x$$, and get something of form $$(a\pm b)^2$$. In this case, since we have a minus sign in front of our second term, we deduce that we want to get something of form $$(a-b)^2$$, such that $$(a-b)^2=a^2-2ab+b^2=x^2-2x+b^2$$ for some $$a,b$$, which we have to figure out. Looking at the first term of this equality, we can clearly let $$a=x$$. Then, $$2x=2ab=2xb$$, so $$b=1$$. Then, $$b^2=1$$. But there is an issue here: here only have $$x^2-2x+y^2=0$$, and we do not have a $$1$$ any where. So we simply add to both sides. Our new equality is $$x^2-2x+1+y^2=1$$ Finally, using the fact that $$(x-1)^2=x^2-2x+1$$, we can conclude that $$(x-1)^2+y^2=1$$

\begin{align}x^2 + y^2 - 2x + 1 = 1 &\leadsto x^2 - 2x + 1 + y^2 = 1 \\ &\leadsto (x - 1)^2 + y^2 = 1\end{align}

$$(x-1)^2 + y^2 = 1$$

Expand to get:

$$(x^2 -2x+1) + y^2=1$$

Arrange the terms to get what you want:

$$x^2 + y^2 = (2x - 1) + 1$$

$$x^2 + y^2 = 2x$$