Factor quadratic expressions including common factor, trinomials, and difference of squares, and use the factored form to reveal zeros (NC.M1.A-SSE.3).

What this topic is asking

NC.M1.A-SSE.3 asks you to write an equivalent factored form of a quadratic $ax^2 + bx + c$ (with $a$ an integer) to reveal the zeros of the related equation or function. Factoring is the workhorse skill of NC Math 1: it powers solving quadratics, graphing parabolas, and reading models. This page covers the main factoring routes you will need.

Step zero: the GCF

Before any pattern, check whether every term shares a factor.

$6x^2 + 9x$ has a common factor of $3x$, so it factors as $3x(2x + 3)$. Pulling out the GCF can also expose a difference of squares ($2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)$) or simplify a trinomial. Skipping this step is the most common reason a factorization is incomplete.

Factoring $x^2 + bx + c$

When the leading coefficient is $1$, factoring is a search for two numbers.

Watch signs: for $x^2 - 7x + 12$, both numbers are negative ($-3$ and $-4$); for $x^2 + x - 12$, the numbers have opposite signs ($+4$ and $-3$).

The difference of squares

$a^2 - b^2 = (a - b)(a + b)$. Recognizable as two perfect squares with a minus sign: $x^2 - 25 = (x - 5)(x + 5)$ and $9x^2 - 16 = (3x - 4)(3x + 4)$. A sum of squares like $x^2 + 25$ does not factor over the reals.

Factoring $ax^2 + bx + c$ by grouping

When $a \ne 1$, factor by grouping.

Factored form reveals the zeros

The reason A-SSE.3 emphasizes factored form is that it shows solutions immediately. If $y = (x - r)(x - s)$, then the zeros are $x = r$ and $x = s$, which are also the $x$-intercepts of the graph. This connects directly to solving quadratic equations by the zero-product property, and it is the inverse of multiplying binomials.

How the NC Math 1 EOC examines this topic

• Multiple choice. Choose the complete factorization, or pick a zero of a function.
• Gridded response. Enter a factor or a zero after factoring.
• Calculator-inactive. Factoring is a core no-calculator fluency skill on the EOC.

Because factoring underpins so many later items, getting it automatic pays off across the Algebra and Functions categories. Treat "factor completely" as a two-part instruction: GCF first, then pattern.

Why factoring and expanding are mirror images

Factoring undoes multiplication. When you expand $(x + 3)(x + 4)$ you get $x^2 + 7x + 12$; when you factor $x^2 + 7x + 12$ you recover $(x + 3)(x + 4)$. Because they are inverse operations, every factorization can be checked by expanding, and the test rewards this habit since factored answers are often scored by exact match. Seeing the two as one reversible process, rather than two unrelated skills, makes both faster and more reliable.

Try this

Q1. Factor $x^2 - 9x + 20$. [1 point]

• Cue. Two numbers multiplying to $20$, adding to $-9$: $-4$ and $-5$, so $(x - 4)(x - 5)$.

Q2. Factor $4x^2 - 25$. [1 point]

• Cue. Difference of squares: $(2x - 5)(2x + 5)$.