Math101 · Independent Ontario learning library
Math101learn.math101.caPerfect Square Trinomials
A perfect square trinomial is produced by squaring a binomial: $a^2+2ab+b^2=(a+b)^2$ or $a^2-2ab+b^2=(a-b)^2$.
Recognizing perfect squares speeds factoring, equation solving, graphing, and completion of squares. The area model explains the otherwise easy-to-forget middle term.
Intuition and core definition
A perfect square trinomial is produced by squaring a binomial: $a^2+2ab+b^2=(a+b)^2$ or $a^2-2ab+b^2=(a-b)^2$. Its first and last terms are squares, and its middle term is exactly twice the product of their square roots, with the sign choosing the binomial.
Notation, language, and conditions
The square on $(a\pm b)^2$ applies to the entire binomial. The leading and constant terms should usually be positive squares over the reals. A GCF must be removed first because a scaled perfect-square pattern may be hidden.
Why this idea matters
A perfect-square trinomial records the expansion of a repeated binomial, with its middle term twice the product of the square roots of the outer terms.
A dependable method
- Factor out any overall GCF.
- Verify the first and last terms are perfect squares.
- Take their square roots as $a$ and $b$.
- Check whether the middle term equals $+2ab$ or $-2ab$.
- Write $(a+b)^2$ or $(a-b)^2$ and expand to verify.
Worked example
Representations and interpretation
An area model for $(a+b)^2$ contains one $a^2$ square, two $ab$ rectangles, and one $b^2$ square. The two identical rectangles explain the coefficient $2$ in the middle term.
Reasoning about variations
A trinomial with square endpoints is not automatically perfect: $x^2+5x+4$ fails because $2(1)(2)=4$, not $5$. It may factor by another trinomial method.
Common mistakes
How to check your work
- Expand the squared binomial.
- Compute $2ab$ independently and compare sign and magnitude.
- Verify the factorization’s leading and constant terms.
Practice
- Factor $x^2+14x+49$.
- Factor $9y^2-24y+16$.
- Is $x^2+10x+36$ a perfect square trinomial?
Answers and brief solutions
Show answers
- $(x+7)^2$ $49=7^2$ and $14x=2(x)(7)$.
- $(3y-4)^2$ The middle is $-2(3y)(4)$.
- No Square root of $36$ is $6$, which would require middle term $12x$.
Synthesis and transfer
Completing a square in a quadratic can be checked by the perfect-square pattern; both the sign and factor of two in the middle term must match.
To rewrite $x^2+10x+25$, take square roots of the outer terms, $x$ and $5$, then verify that twice their product is $10x$. The factorization is $(x+5)^2$. If the middle term were $-10x$, the repeated binomial would use a minus sign; if it were $8x$, the expression would not be a perfect-square trinomial despite having square outer terms. Completing $x^2+10x$ to a square requires adding $25$, which explains why half the linear coefficient is squared. Expansion remains the most direct check of all three coefficients.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Factor $x^2+14x+49$.
- $49=7^2$ and $14x=2(x)(7)$.
End of lesson
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