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Factoring Trinomials Calculator

Enter the coefficients of a quadratic equation to extract the GCF and factor it into linear binomials.

Enter Quadratic Equation

x2
+
x
+

Factored Output

Factored Output

$$ 2(x+1)^2 $$
This is a perfect square!

How to Factor Quadratic Trinomials

Factoring a trinomial of the form $ax^2 + bx + c$ involves breaking it down into the product of two linear binomials, typically formatted as $(px + q)(rx + s)$. Factoring is a crucial skill in algebra, allowing you to easily find the roots (or x-intercepts) of a parabola.

Step 1: Extract the Greatest Common Factor (GCF)

Before splitting the trinomial, check if the coefficients $a$, $b$, and $c$ share a common divisor. If they do, factoring out this GCF makes the subsequent math much easier and guarantees your final binomials are in their simplest integer form.

Example: $2x^2 + 4x + 2$ shares a GCF of 2. Extracting it leaves $2(x^2 + 2x + 1)$.

Step 2: Calculate the Discriminant ($\Delta$)

The discriminant determines whether the quadratic can be factored using integers. It is calculated using the formula:

$$ \Delta = b^2 - 4ac $$
  • If $\Delta < 0$ or is not a perfect square: The trinomial has complex or irrational roots and cannot be factored into binomials with integer coefficients.
  • If $\Delta = 0$: The trinomial is a perfect square. Both binomial factors are identical, resulting in the form $(px + q)^2$.
  • If $\Delta > 0$ and is a perfect square (e.g., 1, 4, 9, 16): The trinomial can be successfully factored into two distinct binomials.

Step 3: Factoring by Grouping (The AC Method)

Once you verify that factoring is possible, multiply the leading coefficient ($a$) by the constant ($c$). Find two integers that multiply to equal $ac$ and add up to equal $b$. You can then split the middle term and factor by grouping.