Factoring Quadratic Trinomials

The factoring method for $x^2 + bx + c$

Factoring a quadratic trinomial means writing it as a product of two binomials: $x^2 + bx + c = (x + p)(x + q)$.

The key question: what two numbers multiply to $c$ and add to $b$?

If you expand $(x + p)(x + q) = x^2 + (p+q)x + pq$, you can see that $p + q = b$ and $p \cdot q = c$.

Step-by-step solution: Factor $x^2 + 7x + 12$

Step 1 — Identify $b$ and $c$. Here $b = 7$ and $c = 12$.

Step 2 — List all factor pairs of $c = 12$:

• $1 \times 12 = 12$, and $1 + 12 = 13$ ✗
• $2 \times 6 = 12$, and $2 + 6 = 8$ ✗
• $3 \times 4 = 12$, and $3 + 4 = 7$ ✓ ← This pair works!

Step 3 — Write the factored form.

$x^2 + 7x + 12 = (x + 3)(x + 4)$

Step 4 — Check by FOIL (First, Outer, Inner, Last):

$(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12 \checkmark$

The area model

The rectangle area model makes this visual: the total area is $x^2 + 7x + 12$. The rectangle has dimensions $(x + 3)$ by $(x + 4)$, and the four sub-rectangles ($x^2$, $3x$, $4x$, $12$) tile the whole rectangle perfectly.

When $c$ is negative

If $c < 0$, the two numbers have opposite signs. For $x^2 + 2x - 15$: need numbers multiplying to $-15$ and adding to $+2$. The pair is $+5$ and $-3$: $(x + 5)(x - 3)$.

When the trinomial can't be factored

Not every trinomial factors with integers. If no factor pair of $c$ adds to $b$, use the quadratic formula or completing the square instead.

Try it in the visualization

Adjust $b$ and $c$. The factor pairs of $c$ are listed with their sums. The pair that adds to $b$ is highlighted. The area model builds a rectangle with matching dimensions. If no pair works, the visualization shows "not factorable over integers."