How to Factor Quadratics

What Does It Mean to Factor a Quadratic?

A quadratic expression looks like \(ax^2 + bx + c\). Factoring means rewriting it as a product of two binomials: \((x + p)(x + q)\). It's the reverse of multiplying with FOIL.

Factoring is used to solve quadratic equations, find zeros of functions, and simplify rational expressions. It's one of the most tested skills in all of high school math.

Step 1 Always: Check for a GCF First

Before anything else, check whether every term shares a common factor. If they do, factor it out first — it simplifies everything that follows.

Example: \(6x^2 + 12x - 18\) → factor out 6 → \(6(x^2 + 2x - 3)\), then factor the trinomial inside.

Case 1: Factoring When \(a = 1\)

When the leading coefficient is 1, the form is \(x^2 + bx + c\). You're looking for two numbers that multiply to \(c\) and add to \(b\).

Example — \(a = 1\)

Factor: \(x^2 + 7x + 12\)

Find two numbers that multiply to 12 and add to 7.
Pairs that multiply to 12: (1,12), (2,6), (3,4)
Which pair adds to 7? → 3 and 4

Write the factored form.
\((x + 3)(x + 4)\)

Check by FOIL:
\(x^2 + 4x + 3x + 12 = x^2 + 7x + 12\) ✓

Example — Negative \(c\)

Factor: \(x^2 + 2x - 15\)

Find two numbers that multiply to −15 and add to 2.
One must be positive, one negative (since \(c\) is negative).
Try: (5, −3) → multiplies to −15 ✓, adds to 2 ✓

Factored form: \((x + 5)(x - 3)\)

Case 2: Factoring When \(a \neq 1\) (AC Method)

When the leading coefficient isn't 1, use the AC method: multiply \(a \cdot c\), find two numbers that multiply to that product and add to \(b\), then split the middle term and factor by grouping.

Example — AC Method

Factor: \(2x^2 + 7x + 3\)

Step 1 — Multiply \(a \cdot c\).
\(2 \times 3 = 6\). Find two numbers that multiply to 6 and add to 7: 6 and 1.

Step 2 — Split the middle term.
\(2x^2 + 6x + 1x + 3\)

Step 3 — Factor by grouping.
\(2x(x + 3) + 1(x + 3)\)
\((2x + 1)(x + 3)\)

Check by FOIL: \(2x^2 + 6x + x + 3 = 2x^2 + 7x + 3\) ✓

Case 3: Difference of Squares

When you have \(a^2 - b^2\), it always factors as \((a + b)(a - b)\). Recognize this pattern and you can factor in one step.

Example — Difference of Squares

Factor: \(x^2 - 25\)

\(x^2 - 25 = x^2 - 5^2 = (x + 5)(x - 5)\)

Both terms must be perfect squares and the sign must be subtraction. A sum like \(x^2 + 25\) does not factor over the real numbers.

Common Mistakes to Avoid

Getting the signs wrong. When \(c\) is positive and \(b\) is negative (like \(x^2 - 5x + 6\)), both factors are negative: \((x-2)(x-3)\). List your factor pairs and check the sum every time.

Skipping the GCF check. Always look for a common factor first. Missing a GCF means extra work later and a wrong final answer if the grader expects fully factored form.

Thinking \(x^2 + c\) factors the same as \(x^2 - c\). It doesn't. Difference of squares requires subtraction. A sum of squares doesn't factor over real numbers.

Not checking with FOIL. Factoring is easy to verify — just multiply your answer back out. Make it a habit.

Factoring Still Feels Like Guessing?

It shouldn't. One session builds the pattern recognition that makes factoring feel automatic. Book a free consultation to get started.

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