<s>[INST] How do I get from $x - x^2 = \frac{1}{4}$ to $x =\frac{1}{2}$? I'm working on a text book problem where I need to sketch the graph of $y = 4x^2 - 4x+1$ by finding where the curve meets the $x$ axis.
To start out I set $y = 0$ then tried to isolate $x$ then,
$4x - 4x^2 = 1$
$x - x^2 =  \frac{1}{4}$
From here I want to continue algebraically to reach $x = \frac{1}{2}$ which I know is the solution from plotting the curve using an app and I can see that that makes sense.
However I am missing some concepts which allow me to turn $x - x^2 = \frac{1}{4}$ into $x = \frac{1}{2}$ and wanted some help to get unblocked.
 [/INST] $$x-x^2 = \frac{1}{4} \implies x^2-x+\frac{1}{4} = 0$$
Here, if you notice, you can see we have two perfect squares: $x^2$ and $\frac{1}{4}$. Also, twice the product of their square roots, which is $2\cdot x\cdot \frac{1}{2}$, gives the coefficient of the middle term. You probably know
$$a^2\pm 2ab+b^2 = (a\pm b)^2$$
Factoring the equation above in the same manner, you get
$$x^2-\color{blue}{x}+\frac{1}{4} = x^2-\color{blue}{2\cdot x\cdot\frac{1}{2}}+\left(\frac{1}{2}\right)^2 = \left(x-\frac{1}{2}\right)^2$$
$$\left(x-\frac{1}{2}\right)^2 = 0$$
$$x-\frac{1}{2} = 0$$
$$x = \frac{1}{2}$$
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