Factoring Quadratics Practice Problems

The Factoring Quadratics lesson organizes every trinomial-factoring technique into one decision process; this practice set drills all of them, asking for one specific constant from the factored form every problem is built backward from.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Leading coefficient of , find the smaller factor	, find
Medium	Difference of squares, or a perfect square trinomial	, find
Hard	The ac method, or a GCF pulled out first	Leading coefficient other than

Easy factors a trinomial with leading coefficient , asking for the smaller of the two constants.

Medium alternates between recognizing a difference of squares and a perfect square trinomial, two patterns worth spotting on sight.

Hard alternates between the ac method for a trinomial whose leading coefficient isn’t , and a trinomial that needs its GCF factored out first before the remaining leading-coefficient-of- trinomial can be factored.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Factor .

Find two numbers that multiply to and add to : and . So , and the smaller constant is .

Medium. Factor and .

The first is a difference of squares: . The second is a perfect square trinomial, because : .

Hard. Factor with the ac method.

. Two numbers that multiply to and add to are and . Split the middle term and group: .

Using the practice problems

Select the difficulty and how many problems you’d like, and the page fills in instantly. Every answer here is a whole number, positive or negative — enter it exactly as computed.

Check Answers scores the set instantly and shows exactly where things went wrong
Show Answers brings up the complete solution key, handy for checking offline attempts
Print Worksheet prepares a print-ready page, suited for classroom use or take-home practice

The rules these practice problems drill

Rule	Statement
Leading coefficient	Two numbers multiplying to , adding to
Difference of squares
Perfect square trinomial
The ac method	Two numbers multiplying to , adding to

See the Factoring Quadratics lesson for the full decision process and worked examples for every case.

Common mistakes to watch for

Skipping the GCF check. A trinomial with a shared numeric factor is far easier once that factor is removed first.

Forgetting the sign rules for the two numbers. A negative always means the two numbers have opposite signs; a positive means they share the sign of .

Misidentifying a perfect square trinomial. The middle term must equal exactly twice the product of the square roots of the first and last terms.

Finding a pair that adds to but doesn’t multiply to . Both conditions are required at the same time in the ac method.

Where to go next

For factoring beyond degree , the Factoring Higher-Degree Practice Problems covers sum and difference of cubes and quadratic-in-form quartics. For the full decision process and every pattern, see the Factoring Quadratics lesson.

Frequently Asked Questions

Why does this practice set ask for p or q instead of the full factored form?+

A fully factored trinomial like (x+3)(x+4) is an expression, which can't be automatically checked as a single number. Every problem is built backward from a specific factored form, so asking for one of its constants is a reliable check on the underlying factoring skill.

How do I know which technique applies to a given trinomial?+

Check the leading coefficient first: if it's 1, find two numbers that multiply to c and add to b. If it isn't 1, use the ac method. Separately, check whether the trinomial matches the difference of squares or perfect square pattern, which factors even faster.

Why do Medium problems sometimes ask about a pattern instead of two numbers?+

Recognizing the difference of squares or a perfect square trinomial on sight is faster than searching for two numbers, and it's worth practicing as its own skill rather than only as a special case of the general method.

How is this different from the Factoring Practice Problems?+

The Factoring practice set mixes several factoring situations together, including a GCF-first case. This practice set is dedicated entirely to trinomials — every leading-coefficient case and both special patterns — for more focused repetition.

Do I need Factoring by Grouping before this practice set?+

The ac-method problems here use exactly the grouping technique from that lesson, so it helps to be comfortable with grouping first, though this practice set can still be used to see where that technique gets applied.

Is there a maximum number of sets I can create?+

There isn't one. Sets are produced on demand, permanently free, with no registration required.

x²+Factoring Quadratics Practice Problems