Fourth Powers

x^4 - 17x^2 + 16 = 0 looks like a hard equation of degree four, but it only holds x^4 and x^2, and x^4 is (x^2)^2. Read x^2 as one unknown and it is a quadratic: that unknown squared, minus 17 of it, plus 16. Two numbers that multiply to 16 and add to -17 are -1 and -16, so it factors as (x^2 - 1)(x^2 - 16) = 0.

Each bracket is a difference of squares, so the left side breaks into four factors: (x + 1)(x - 1)(x + 4)(x - 4). Split into cases and you get four answers: 1, -1, 4 and -4. An equation of degree four has at most four solutions, and this one has all of them.

Not every bracket gives real answers. x^4 - 21x^2 - 100 = 0 factors as (x^2 - 25)(x^2 + 4) = 0, and x^2 = -4 has no real solution, since the square of a real number is never negative. The only real solutions are 5 and -5.