factoring to solve

Factoring to solve is the fastest route to a quadratic's roots when the expression breaks cleanly into a product of simpler pieces. The idea rests on one bedrock fact: if a product of numbers equals zero, at least one of the factors must be zero. So you split the problem into easy linear ones.

The recipe has three steps. First, move everything to one side so the equation reads (expression) = 0. Second, factor the expression into a product, like (x - r)(x - s). Third, apply the zero-product property: set each factor equal to zero and solve, giving x = r and x = s.

The catch is that not every quadratic factors nicely over the integers — many have irrational or complex roots that resist factoring by hand. The discriminant tells you in advance: if b^2 - 4ac is a perfect square (with integer coefficients), factoring will work; otherwise reach for completing the square or the quadratic formula. Also, both sides being equal to zero matters — you cannot factor and read off roots unless the equation is set to zero first.