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Solve: x^{2} + 2x + 1 = 16.

We can factor the left-hand side of this equation to see that:

(x + 1)^{2} = 16

Then snag the square root of both sides:

x + 1 = ±4

Solve the two equations:

x + 1 = 4 and x + 1 = -4

And our solutions are x = 3 and x = -5. Phew. It's nice to have some closure on that.

Solve: x^{2} = 5.

The answers are , and there's no prettier way to write them. We could inscribe them inside a drawing of a floral wreath, but that's about it.

Solve (x – 6)^{2} = 15.

We take the square root to find:

So we need to solve these two equations:

The solutions are, respectively:

That's as nice as they get. Getting uglier. Must...look...away...

Solve -2x^{2} = 16.

We divide both sides by -2 to find x^{2} = -8.

Now we're stuck, without any butter or Vaseline handy. There is no real number whose square is -8. This equation has no real number solutions. Unreal.

Make it rain.

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