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Q:

iOS 8 Custom Calendar Event

I am currently working on an app that must support iOS 7 and above. I know that Apple has removed the existing EventKit API, so I found a script that can be used to setup a custom calendar view that will work for iOS 7 and above.
However, I need to be able to send notifications for events to the user and that is not possible with Apple’s new calendar widget. I am considering doing something custom that uses private APIs to create events in Calendar. Please provide feedback on this approach.
Thanks!

A:

Apple has added the ability to send notifications directly from the Calendar app. If you are running iOS 8 it will automatically generate a notification when you add an event.

In the Calendar app, open the Settings app and then open Notifications under the Messages section. Under “calendar” choose “Check for new events” and then choose “allow” under “give me notifications for:…”
This will show you the new events in the Calendar app.

Q:

How to prove -1 is the only number satisfying $x^2 + x + 1 = 0$

How can you prove that -1 is the only number that satisfies
$$x^2 + x + 1 = 0.$$
I know that if $x = 0$, the equation becomes $1 = 0$, and so $x = -1$. However, how can I prove that no other numbers satisfy the equation?

A:

You may want to use the case $x=-1$ as the base case:
For $x e -1$ note that $x^2+x+1=0$ implies
$$x^2+x=\frac12 (x^2+x+1)=\frac12(x+1)^2$$
and so $x+1=\pm\sqrt2$ and
$$(x+1)^2=4\quad\Longrightarrow\quad x^2+x+1=0\quad\Longleftrightarrow\quad x=\frac{ -1\pm\sqrt2}2$$

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