<s>[INST] Explain why there's no solution to the equation $2x-2x^2 = 1$ How can you tell there's no solution to the equation $2x - 2x^2 = 1$.
The supporting information goes like this:
The diagram above shows the graph of $y = 7 + 2x - 2x^2$.
I tried to do this:
$2 \cdot 1^2 = 2y$.
therefore $y = 2-2 = 0$;
 [/INST] We want to solve $2x^2-2x+1=0$, presumably in the reals. Look at the upward-facing parabola $y=2x^2-2x+1$. 
Complete the square. We have $x^2-x=(x-1/2)^2-1/4$. Thus our equation can be written as 
$$y=2(x-1/2)^2+1/2.$$
The vertex of the parabola is at $x=1/2$. There, $y=1/2$, so the parabola is always above the $x$-axis. Thus it has no point in common with the $x$-axis, and therefore our equation has no real solutions.
The arithmetic is a little easier if we look at the equivalent equation $4x^2-4x+2=0$, which can be rewritten as $(2x-1)^2+1=0$. But $(2x-1)^2$ is always $\ge 0$ if $x$ is real. So we can't have $(2x-1)^2+1=0$ for real $x$.
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