How to Factor a Perfect Square Trinomial

Algebra 2 · Factoring polynomials

How to Factor a Perfect Square Trinomial

Problem

x^2 + 10x + 25

Answer

(x + 5)^2

A perfect square trinomial fits the pattern a squared plus or minus 2ab plus b squared. It always factors into one bracket squared.

Step-by-step solution

Check that the first and last terms are perfect squares.
$x^2 = (x)^2,\quad 25 = (5)^2$
Why: A perfect square trinomial starts and ends with squares.
Check the middle term against 2ab.
$2 \cdot x \cdot 5 = 10x$
Why: The middle term must be twice the product of the two square roots.
Write the square of a binomial.
$(x + 5)^2$
Why: Both signs are plus because the middle term is positive.

Common mistakes

Writing (x + 5)(x - 5). That is a difference of squares and has no middle term.
Forgetting the factor 2 in 2ab and checking only x times 5.
Missing the sign. A negative middle term gives (x - 5)^2.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$x^2 + 6x + 9$

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$(x + 3)^2$
$x^2 - 8x + 16$

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$(x - 4)^2$
$4x^2 + 12x + 9$

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$(2x + 3)^2$
$x^2 - 14x + 49$

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$(x - 7)^2$
$9x^2 - 6x + 1$

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$(3x - 1)^2$