Completing the Square and the Quadratic Formula

The quadratic formula is simply a recipe that solves every quadratic equation, no factoring or cleverness required. It looks like the scariest thing in algebra, but it's just a few pieces: $a$, $b$ and $c$ (the three numbers in your equation), $-b$ and $2a$ (where the answer is centered), and $\pm\sqrt{b^2 - 4ac}$ (how far the two answers sit on either side of that center). The part under the root, $b^2 - 4ac$, is called the discriminant because it discriminates, or tells apart, the three possible cases. Bear with it; we'll see it's just completing the square, done once and for all.

The problem with factoring. Last lesson, factoring worked because the numbers were friendly. Try $2x^2 - 4x - 3 = 0$ and no nice factors exist. So what can we do? We noticed then that equations like $(x - 2)^2 = 9$ are easy, because you just take a square root. The idea here is to force any quadratic into that shape.

Completing the square. Notice that $(x + 3)^2 = x^2 + 6x + 9$: the 9 is half of 6, squared. So for $x^2 + bx$, adding $\left(\frac b2\right)^2$ always makes a perfect square:

To solve $x^2 + 6x - 7 = 0$, move the 7 over and add 9 to both sides: $x^2 + 6x + 9 = 16$, so $(x + 3)^2 = 16$. Now take square roots: $x + 3 = \pm4$, so $x = 1$ or $-7$. There's a video further down showing why this is literally completing a square.

The quadratic formula is what you get by completing the square on $ax^2 + bx + c = 0$ with the letters left in. It works every time, including when factoring is hopeless.

The discriminant predicts the answer before you compute it. If $b^2 - 4ac > 0$, the $\pm$ gives two different real roots. If it's $0$, the $\pm$ adds nothing and there's one. If it's negative, you'd need the square root of a negative number, so the roots are complex, and the parabola never crosses the x-axis.

The quadratic formula gives the time a ball hits the ground, the price that maximizes revenue, and the resonant frequency of a circuit.