Once you can factorise a quadratic, you can solve one. To solve x^2 + bx + c = 0, first rewrite the left-hand side as a product of two brackets:

Now the whole expression is two things multiplied together, and that product equals zero. This is where one clean idea does all the work — the null-factor law: if two numbers multiply to give 0, then at least one of them must be 0. (No two non-zero numbers can multiply to zero.) So:

Each of those is a simple equation giving one solution: x = -p or x = -q. A quadratic usually has two solutions, called its roots.

Take x^2 + 7x + 12 = 0 We need two numbers that multiply to 12 and add to 7 — that is 3 and 4. So it factorises and then splits into two pieces:

Solving each piece gives the two roots x = -3 or x = -4. Watch each stage build.

A neat special case: when c is a perfect square subtracted, like x^2 - 9 = 0, the factors are a difference of two squares, (x - 3)(x + 3) = 0, giving roots x = 3 or x = -3. The same null-factor law finishes the job.

See it explained

Sal Khan works a quadratic equation from factorised form to its roots.