Factor x² – 8x + 16.

Like a goddess, or an Adonis, this trinomial is perfect. The first and last terms are perfect squares (of x and 4), and the middle term equals twice the product of those, or 2(x)(4).

This factors to a squared binomial. Take the square root of the first term and the square root of the last term and throw a "–" sign in between them.

x² – 8x + 16

= (x – 4)²

Factor 9x² + 42x + 49.

We can see that the first and last terms are perfect squares, of 3x and 7. The middle term also passes the test: 42x = 2(3x)(7).

This is a perfect square trinomial. Factor it to a squared binomial.

9x² + 42x + 49

= (3x + 7)²

The binomial has a "+" sign in it because the middle term of the trinomial is positive.

Factor 25x⁴ – 20x² + 4.

We have plenty of perfect squares, and our middle term fits the prophecy perfectly: 20x² = 2(5x²)(2). The coming of The Great Weasel will happen soon, as was fortold.

Oh, what? This is a perfect square trinomial, which factors to a squared binomial? Well, don't we feel foolish.

25x⁴ – 20x² + 4

= (5x² – 2)²

How about we all move on, while we go in the back and put away the robes and candles.

Solve x² + 16x + 64 = 0.

First factor the equation, then solve the equation for its roots.

x² + 16x + 64 = (x + 8)²

(x + 8)² = 0

x = -8

Just like that.

Factor x² – 25.

This is a difference of squares. It will factor to a binomial sum and a binomial difference.

Just take the square root of both terms and write them as the product of a sum and difference.

(x + 5)(x – 5)

Factor 4x² – 1.

Take the positive square root of the first and last terms and write them as a binomial sum and a binomial difference.

4x² – 1 = (2x – 1)(2x + 1)

Solve 16x² – 49 = 0.

The square roots of the two terms are 4x and 7. Use these to factor the difference of the two squares, then solve.

(4x + 7)(4x – 7) = 0

4x + 7 = 0 and/or 4x – 7 = 0

Solve 25x² – 25 = 0.

This problem can be made a little easier by factoring out the 25 first.

25(x² – 1) = 0

The remaining term is (also) a difference of two squares.

25(x + 1)(x – 1) = 0

x = -1, 1

Math shmarter, not harder.