Quadratic Formula

1. Introduction: One Formula That Solves Every Quadratic

You have already met quadratic equations. They look like $ax^2 + bx + c = 0$, where the highest power of $x$ is 2. The letter $a$ is the number in front of $x^2$, $b$ is the number in front of $x$, and $c$ is the number standing alone. There is one rule: $a$ can never be 0, because then there would be no $x^2$ and it would not be a quadratic at all.

You have also solved some of these by factorising — by splitting the middle term and pulling out two brackets. That method is fast when the numbers are friendly. But here is the part that trips people: sometimes the two numbers just refuse to come. You hunt and hunt, and nothing multiplies and adds the way you need.

So you need a tool that works every single time, no matter how ugly the numbers are. That tool is the quadratic formula. Feed it $a$, $b$, $c$, and it hands you the answers.

And there is a bonus. Buried inside that formula is one small expression called the discriminant. It lets you look at a quadratic and say — before you do any real work — how many real answers it even has. One, two, or none. That is a quiet superpower.

This whole lesson is about those two things: the formula that always works, and the discriminant that tells the future.

You can now say why a universal method is needed and name the two ideas this lesson teaches.