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December 23rd, 2017, 03:49 PM   #11
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The problem doesn't say so. However, let's treat the current ungrammatical wording ("where x-intercept") as though it were "where the only $x$-intercept".

It's now easy to find examples that show that the statements numbered 1, 2, and 4 are incorrect. With a little more trouble (because 9/8 is fairly close to, but less than, $\frac{2\sqrt3}{3}$), one can also choose a value of $a$ such that the statement numbered 4 is incorrect.

The statement numbered 5 is correct. This can be proved in various ways. It's equivalent to the condition $4 < 27b^2\!$.

Note that your early assertion that both factors must equate to zero is incorrect. For there to be only one real zero, the discriminant of the second factor must be negative (so that it has no real zero).
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