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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, taking the square root of 16 yields

x + 1 = ± 4.

Solve the two equations,

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

and we can conclude that the 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 a square root to find

,

so we need to solve the two equations

.

The solutions are, respectively,

Those are the solutions, and 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.