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Perfect Square Trinomial

A perfect square trinomial is a three-term polynomial that can be written as the square of a binomial. It follows the pattern a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2 or a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2.

A perfect square trinomial is a quadratic expression of the form a2±2ab+b2a^2 \pm 2ab + b^2, where aa and bb represent any real-valued expressions. It is the result of squaring the binomial (a±b)(a \pm b), and it factors back into (a±b)2(a \pm b)^2. The key identifying feature is that the middle term equals exactly twice the product of the square roots of the first and last terms.

Key Formula

a2+2ab+b2=(a+b)2a2−2ab+b2=(a−b)2\begin{gathered}a^2 + 2ab + b^2 = (a + b)^2\\a^2 - 2ab + b^2 = (a - b)^2\end{gathered}
Where:
  • aa = the square root of the first term of the trinomial
  • bb = the square root of the last term of the trinomial
  • 2ab2ab = the middle term, equal to twice the product of a and b

Worked Example

Problem: Factor the trinomial x2+10x+25x^2 + 10x + 25.
Step 1: Check whether the first and last terms are perfect squares.
x2=(x)2  ✓and25=(5)2  ✓x^2 = (x)^2 \;\checkmark and 25 = (5)^2 \;\checkmark
Step 2: Identify aa and bb from those square roots.
a=x,b=5a = x, \quad b = 5
Step 3: Verify that the middle term equals 2ab2ab.
2ab=2(x)(5)=10x  ✓— thismatchesthemiddleterm2ab = 2(x)(5) = 10x \;\checkmark \quad\text{— }this matches the middle term
Step 4: Since the middle term is positive, write the factored form using the addition pattern.
x2+10x+25=(x+5)2x^2 + 10x + 25 = (x + 5)^2
Answer: x2+10x+25=(x+5)2x^2 + 10x + 25 = (x + 5)^2

Why It Matters

Recognizing perfect square trinomials speeds up factoring significantly — instead of testing factor pairs, you apply the pattern directly. This skill is essential for completing the square, which is how you derive the quadratic formula and convert quadratic functions into vertex form. It also appears frequently when simplifying radical expressions and solving equations in physics and engineering.

Common Mistakes

Mistake: Forgetting to check the middle term and assuming any trinomial with two perfect square terms is a perfect square trinomial.
Correction: Always verify that the middle term equals 2ab2ab. For example, x2+7x+9x^2 + 7x + 9 has perfect square first and last terms, but 2(x)(3)=6x≠7x2(x)(3) = 6x \neq 7x, so it is not a perfect square trinomial.
Mistake: Writing (a+b)2(a + b)^2 when the middle term is negative.
Correction: If the middle term is negative, the factored form uses subtraction: a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2. The sign of the middle term determines the sign inside the binomial.

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