The Quadratic Formula, Demystified

Every quadratic $ax^2 + bx + c = 0$ is solved by one formula:

It looks like something to memorize and fear. It's actually just completing the square performed on the general equation — once — so you never have to do it again.

The discriminant tells the story first

The piece under the root, $b^2 - 4ac$, is the discriminant. Before solving, it tells you how many real solutions exist:

• $b^2 - 4ac > 0$: two real roots (the parabola crosses the $x$-axis twice).
• $b^2 - 4ac = 0$: one repeated root (it just touches the axis).
• $b^2 - 4ac < 0$: no real roots (it floats above or below the axis).

A worked example

Solve $2x^2 - 4x - 6 = 0$. Here $a = 2,\ b = -4,\ c = -6$.

So $x = 3$ or $x = -1$. Always substitute one back to check — $2(9) - 12 - 6 = 0$. ✓

Tip

Compute the discriminant first. If it's negative, you can stop — there are no real solutions, and you've saved yourself the rest of the arithmetic.