Key takeaway: A perfect square trinomial is a three-term expression that results from squaring a binomial, so it factors as (a + b)² or (a − b)².
Grade 9 – Algebra 1A-SSE.A.2Expressions and Equations
Definition of Perfect Square Trinomial
A perfect square trinomial is what you get when a binomial is multiplied by itself. Its first and last terms are perfect squares, and its middle term is exactly twice the product of the square roots of those two terms.
Test an expression like \(4x^{2} + 28x + 49\) in three steps. First, \(4x^{2} = (2x)^{2}\) and \(49 = 7^{2}\). Second, twice the product of \(2x\) and 7 is \(2 \cdot 2x \cdot 7 = 28x\), which matches the middle term. Third, write the square: \((2x + 7)^{2}\). If the middle term is negative, the binomial has a minus sign. Recognizing the pattern turns factoring into a quick check.
Formula and notation
Two patterns, one with each sign:
\[a^{2} + 2ab + b^{2} = (a + b)^{2} \qquad a^{2} – 2ab + b^{2} = (a – b)^{2}\]
The last term is always positive, because it is a square.
Key facts
- The first and last terms are squares, and the last term is positive.
- The middle term must equal twice the product of the two square roots.
- The sign of the middle term gives the sign inside the binomial.
- If the middle term fails the check, the trinomial needs another factoring method.
Perfect Square Trinomial example
Decide whether \(4x^{2} + 28x + 49\) is a perfect square trinomial, and factor it if so.
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Square roots of the end terms: \(\sqrt{4x^{2}} = 2x\) and \(\sqrt{49} = 7\).
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Middle term check: \(2 \cdot (2x) \cdot 7 = 28x\), which matches.
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Write the square of a binomial: \((2x + 7)^{2}\). Check: \(4x^{2} + 14x + 14x + 49 = 4x^{2} + 28x + 49\).
Yes, and it factors as \((2x + 7)^{2}\).
Common mistake
Watch out
Ben squares the binomial term by term: \((2x + 7)^{2} = 4x^{2} + 49\).
Why it is wrong
Squaring a binomial means multiplying it by itself, \((2x + 7)(2x + 7)\), so the two cross products appear: \(14x + 14x = 28x\). The correct expansion is \(4x^{2} + 28x + 49\); the middle term cannot be dropped.
Related terms and lessons
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