MHB Can Plugging in (x/2) or (x - 1/2) Determine Real Zeros in a Quadratic Equation?

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I can replace f(x) with x - x^2. Should I plug (x/2) into f(x)? How about (x - 1/2) into f(x)? I need the set up.

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Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?
 
Greg said:
Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?

Let y = the function.

y = x - x^2 - 1/2 does not cross the line y = 0.
 
Note the concavity of $f(x)$. What does that tell you about lowering the graph of $f(x)$? (the line $y=0$ does not concern us presently).
 
write each quadratic in standard form, $ax^2+bx+c$ ... check each discriminant, $D = b^2-4ac$

you know what the discriminant can tell you about the nature of zeros, right?
 
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I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
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