Recognize and factor perfect square trinomials

Key concept: Perfect square trinomials follow a very specific pattern. If the first and last terms are squares and the middle term matches twice the product of their roots, the factorization is straightforward.

Step by step

A trinomial is a perfect square if it matches

$A^2 \pm 2AB + B^2 = (A \pm B)^2$

a)

$x^2+14x+49=(x+7)^2$

b)

$k^2+8k+16=(k+4)^2$

c)

$y^2-10y+25=(y-5)^2$

d)

$25y^2-70y+49=(5y-7)^2$

e)

$36c^2-108c+81=(6c-9)^2$

f)

$121+176y+64y^2=(11+8y)^2=(8y+11)^2$

To verify, expand each bracket and check that the middle term matches exactly.

Pitfall alert

The middle term is where most errors happen. If the sign or coefficient is off by even 1, the trinomial is not a perfect square. It helps to test the square roots of the first and last terms first, then compare the middle term to $2AB$.

Try different conditions

If the last term were not a square number, the trinomial would usually not factor as a perfect square. For instance, $x^2+14x+48$ cannot be written as $(x+a)^2$ because $48$ is not a square and the middle term does not fit the pattern.

Further reading

perfect square trinomial, binomial square, factoring