Factoring Binomials (Perfect Squares)

The three-part check

Take $x^{2} + 10x + 25$. First term: $x^{2}$ is the square of $x$. Last term: $25 = 5^{2}$. Middle term: $2 \cdot x \cdot 5 = 10x$ — matches. All three checks pass, so it factors as $(x + 5)^{2}$.

The middle-term check is the one that actually decides. $x^{2} + 11x + 25$ has perfect squares on both ends, but $11x \neq 2 \cdot x \cdot 5$, so it is not a perfect square and needs a different method (in this case, it does not factor nicely at all).

The sign of the middle term picks the sign in the answer: $+2ab$ gives $(a + b)^{2}$ and $-2ab$ gives $(a - b)^{2}$. The last term is positive either way — if the constant is negative, the trinomial is not a perfect square, period.

The area model below builds $(x + 3)^{2}$ as a square with side $x + 3$. The two off-diagonal rectangles are each $3x$, and together they make the middle term $6x = 2 \cdot x \cdot 3$ — which is exactly why the pattern's middle term is $2ab$.

Reading off the answer

Once the pattern checks out, the factored form uses the two square roots: the square root of the first term and the square root of the last term, joined by the middle term's sign, all squared. For $x^{2} - 8x + 16$: roots are $x$ and $4$, the middle sign is minus, so the answer is $(x - 4)^{2}$.

Writing $(x - 4)^{2}$ or $(x - 4)(x - 4)$ are both correct — the exponent form is just tidier.

Leading coefficients and why the pattern is worth spotting

The pattern extends to trinomials like $9x^{2} + 30x + 25$: the first term is $(3x)^{2}$, the last is $5^{2}$, and the middle is $2 \cdot 3x \cdot 5 = 30x$. It factors to $(3x + 5)^{2}$.

You could factor these by grouping instead, and you would get the same answer with more work. Recognizing the perfect-square shape saves time now and becomes essential later — completing the square and vertex form in the quadratics unit are built directly on this pattern.

Worked examples

Try one yourself

Common questions