Perfect Square Trinomial
In short: Perfect square trinomial describes a three-term expression like x² + 10x + 25 that equals a binomial multiplied by itself, here (x + 5)².
Algebra 1AlgebraA-SSE.A.2
What is Perfect Square Trinomial?
Multiply a binomial by itself and what comes out is a perfect square trinomial: \((x + 5)^2 = x^2 + 10x + 25\). Two of its terms are squares, and the remaining one is double the product of their roots.
Two patterns cover every case: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 – 2ab + b^2 = (a – b)^2\). The middle sign matches the sign inside the binomial, while the final term is always positive.
How to spot it
- The two end terms are squares, such as \(4x^2\) and 25, and the last one is positive.
- The middle term seems to be twice the product of the two square roots.
- A task wants a quadratic solved by recognizing a squared binomial.
- A direction asks which constant turns \(x^2 + 14x\) into a square.
The method
- Check that the first and last terms are perfect squares and that the last is positive.
- Take their square roots to get a and b.
- Test whether the middle term equals \(2ab\) in size.
- Write \((a + b)^2\) for a plus sign in the middle, or \((a – b)^2\) for a minus sign.
Perfect Square Trinomial: solved example
Problem
Test whether \(4x^2 + 20x + 25\) fits the squared-binomial pattern, and factor it when it does.
Solution
The first term is \((2x)^2\) and the last is \(5^2\), so \(a = 2x\) and \(b = 5\).
The middle term should be \(2ab = 2(2x)(5) = 20x\), which matches. The expression is a perfect square trinomial and equals \((2x + 5)^2\).
Try it
Question 1
Factor \(x^2 – 12x + 36\).
Show the answer
Here \(a = x\) and \(b = 6\), and the middle is \(-2 \cdot x \cdot 6 = -12x\), so \((x – 6)^2\).
Question 2
Which constant c completes \(x^2 + 14x + c\) into a square?
Show the answer
\(c = \left(\frac{14}{2}\right)^2 = 49\), giving \((x + 7)^2\).
Perfect Square Trinomial vs. Difference of Squares
Squaring a binomial leaves a middle term. Multiplying conjugates, \((a + b)(a – b)\), leaves none. A frequent slip is writing \((x + 5)^2 = x^2 + 25\), which drops the middle \(10x\) completely.
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Whole dictionary: A–Z · Algebra 1 · Algebra