Solving quadratics by factoring
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Solve x^2 - 5x + 6 = 0.
Common mix-up
In this example, a student wrote x = 2.
The student solved only the first factor, x - 2 = 0, and stopped at x = 2, never setting the second factor to zero.
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See why this works
Why does x^2 - 5x + 6 = 0 have two answer boxes to fill?
x^2 - 5x + 6 = 0 has two solutions, not one. Once the quadratic is factored, a product equal to zero tells you exactly where they hide.
If a product is zero, at least one of its factors must be zero.
A quadratic equation usually has two solutions, and factoring exposes both of them. Start with x^2 - 5x + 6 = 0. Looking for a pair with product 6 and sum -5 gives -2 and -3, so the equation becomes (x - 2)(x - 3) = 0. Nothing about the solutions is visible yet, but the product form is ready.
The zero product property is the reason two answers appear. The equation says a product equals zero. A product can only be zero when at least one of the factors is zero. Numbers are not sneaky here: if two things multiply to nothing, one of them must itself be nothing.
So split the single equation into two separate ones. Set x - 2 = 0, which gives x = 2. Then set x - 3 = 0, which gives x = 3. Both values solve the original equation, and each one makes exactly one factor zero while the other factor takes care of itself.
Substitute both answers back to be sure. At x = 2, the equation reads 4 - 10 + 6 = 0. At x = 3, it reads 9 - 15 + 6 = 0. Finding one factor and stopping is the most common error, and the check is the fastest way to notice that the equation still has a second solution waiting.
Keep this idea
Set each factor equal to zero and solve it. Every factor contributes one solution, and every solution must check in the original equation.
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