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If a rocket launches into the air with height modeled by h=−16t2+40th = -16t^2 + 40t, taking a square root won't isolate tt because of that stubborn middle term.
To find when the rocket touches the ground (h=0h=0), we need a way to turn addition into multiplication.
What magic property lets us split a single equation into two simpler ones?

The Zero-Product Property

The Zero Product Property states that if A⋅B=0A \cdot B = 0, then A=0A = 0, B=0B = 0, or both are 00. This allows each linear factor to be set independently equal to zero.
In 1631, English mathematician Thomas Harriot first realized that setting an entire polynomial equation equal to zero turns complicated algebra into simple factor-by-factor solutions.
This property guarantees that any factorable quadratic yields rational roots — solutions that can be written as clean fractions or whole numbers pq\frac{p}{q}.
How do we rewrite an expression like x2−5x+6x^2 - 5x + 6 as a product of two linear factors?

Factoring Monic Quadratics (a=1a = 1)

A monic quadratic is simply a quadratic expression where the leading coefficient aa equals 11, written as x2+bx+cx^2 + bx + c.
Because (x+p)(x+q)=x2+(p+q)x+pq(x + p)(x + q) = x^2 + (p + q)x + pq, we search for two integers that multiply to give cc and add up to give bb.