AI Tutor
If two mystery numbers multiply together to give zero, what can you say for sure about at least one of them?
One of them must be zero. This simple insight turns out to be the secret key to solving quadratic equations.

The Zero Product Property

The zero product property (also called the null factor law) states that if A×B=0A \times B = 0, then either A=0A = 0, B=0B = 0, or both are zero.
Mathematician Thomas Harriot first popularized setting polynomial equations to zero in the early 1600s, realizing it made them vastly easier to break down.
How do we rewrite an equation with an x2x^2 term into two multiplying factors? Let's break down the mechanics.

Factoring Monic Quadratics

A monic quadratic is an expression of the form x2+bx+cx^2 + bx + c, where the leading coefficient (the number multiplying x2x^2) is simply 11.
To factor x2+bx+cx^2 + bx + c into (x+p)(x+q)(x + p)(x + q), we need two integers pp and qq that multiply to cc and add up to bb.
This works because expanding (x+p)(x+q)(x + p)(x + q) gives x2+(p+q)x+pqx^2 + (p + q)x + pq, directly matching our original expression.
Now that we know how to factor, how do we find the actual solutions to an equation?