Solve: x2 + 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: x2 = 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 -2x2 = 16.
We divide both sides by -2 to find x2 = -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.