Factor quadratics completely, apply the zero-product property, and solve every factor equation.
This test has 20 questions
After the test · factor-to-root workshop
This practice set focuses on solving rather than merely recognizing a factorization. The equation may be monic or nonmonic, may contain a greatest common factor, may hide a difference of squares, may produce a repeated root, or may need to be moved to standard zero form before any factoring begins. The finish line is not the factored expression — it is the complete set of solutions.
Master solving route
Factoring methods work cleanly when the equation is written as expression = 0.
Do not skip a common factor just because the remaining trinomial also factors.
This is the zero-product property.
Collect all distinct roots, including x = 0 when x is a factor.
Quick factoring shelf
AB = 0 → A = 0 or B = 0
Each factor is a separate route to a root.
ax² + bx = x(ax + b)
If x is factored out, x = 0 is one possible root.
a² − b² = (a − b)(a + b)
This pattern produces two linear factors immediately.
(x − r)² = 0 → x = r
The factor appears twice, but the solution value is the same.
(x + m)(x + n), where m + n = b and mn = c
Look for two numbers with the needed sum and product.
(px + q)(rx + s)
The leading coefficients and constant factors must multiply correctly while the middle terms combine to b.
Visual factor tree
Move to zero first
The zero-product property cannot yet be applied.
Now each factor can be set equal to zero.
Monic versus nonmonic trinomials
Find two numbers that add to 7 and multiply to 12, then solve both factor equations.
The leading coefficient must be distributed across the factors while the middle terms still combine correctly.
Difference of squares
Both terms are perfect squares and they are separated by subtraction.
The identity a² − b² = (a − b)(a + b) gives the factors immediately.
Repeated roots
The same linear factor occurs twice.
The multiplicity is 2, but there is only one distinct root.
Do not stop at the factorization
This is the central emphasis of the page: factoring is a tool for producing simpler equations. The final response must be the roots, not merely a correct product of factors.
(x − 2)(x + 6) = 0 is not yet the finished answer.
x − 2 = 0 and x + 6 = 0.
x = 2 and x = −6.
Common mistakes from the page
The zero-product property requires a product equal to zero.
The common factor may itself generate a valid root.
The page explicitly focuses on solving rather than merely identifying a factorization.
Every factor must be set equal to zero.
The factor has multiplicity two, but the value is the same.
Final factoring checklist