The Formula, Stated

Some quadratics resist the machine, so mathematics built a master key: it is what comes out of forcing any quadratic into a squared package, a walk a later section takes step by step. For any equation ax^2 + bx + c = 0, the solutions are given by the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Watch it work on x^2 + 5x + 6 = 0, where a = 1, b = 5, c = 6. Substitute: x = (-5 ± √(5^2 - 4 * 1 * 6)) / (2 * 1). Work the inside: 25 - 24 = 1, and √1 = 1, so x = (-5 ± 1) / 2: the two cases give -2 and -3, the same pair factoring finds.

The formula is not magic to admire but a procedure to run: substitute a, b and c, work the arithmetic under the root, take the root, split the ±, finish each case. On this lesson's board the factoring chips are off; the formula carries you the whole way.