Solve quadratic equations by the factor method

Follow this one with me, a step at a time. Try to guess each step first.

Worked example 1

Solve x² − 21x + 108 = 0 by the factor method. The roots are listed in ascending order.

Remember: Look for two numbers with product 108 and sum −21.

−9 and −12 have product 108 and sum −21, so (x − 9)(x − 12) = 0.
One of the factors must be zero, so x − 9 = 0 or x − 12 = 0.
The roots are x = 9, 12.

Common mistakes

Used the numbers in the factors as the roots. If (x − a) is a factor then x = a, with the sign changed.
One root has the wrong sign. Substitute each root into the equation to check.
These two numbers give the right sum and the wrong product. Both have to fit.