b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

The quadratic equation is essentially a shortcut to completing the square. It may not seem like a shortcut to your students, but it beats spending an hour trying to factor an equation that just won't budge.

Students should know the quadratic formula and how to use it. When given a quadratic equation ax² + bx + c = 0, the quadratic formula is this ravishing thing:

Of course, if all that business under the radical turns out to be negative, we've got a real problem. We can't square root negative numbers, right? As long as students are familiar with complex numbers and i, this real problem becomes an imaginary one very quickly. Then we can simplify the rest so that it follows the form a ± bi.

While students may cringe at this monstrous behemoth of a formula, they'll soon appreciate its sophisticated and simple beauty. In fact, many of them will memorize it without even realizing it. Armed with those three ways, students should be able to attack just about any quadratic formula using only a pencil and their bright and shiny noggins (and maybe a calculator, if you're feeling generous).