How to Solve Perfect Square Trinomial?
A perfect square trinomial is the result of squaring a binomial, and recognizing the pattern lets you factor or expand these expressions in one quick step. Understanding perfect square trinomials is essential for completing the square, solving quadratic equations, and simplifying many algebraic expressions. This guide covers both formulas, how to spot them, and plenty of practice.
Solve Perfect Square Trinomial: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Solve Perfect Square Trinomial: pop-up practice
What Is a Perfect Square Trinomial?
A perfect square trinomial is a trinomial that equals the square of a binomial. It comes in two forms:
- \(\color{blue}{(a + b)^{2} = a^{2} + 2\text{ ab } + b^{2}}\)
- \(\color{blue}{(a – b)^{2} = a^{2} – 2\text{ ab } + b^{2}}\)
The first and last terms are perfect squares, and the middle term is exactly twice the product of their square roots.
How to Recognize a Perfect Square Trinomial
The Three-Step Check
- The first term must be a perfect square: \(\color{blue}{a^{2}}\).
- The last term must be a positive perfect square: \(\color{blue}{b^{2}}\).
- The middle term must equal \(\color{blue}{\pm 2\text{ ab }}\).
If all three conditions are met, the trinomial is a perfect square.
Recognition Examples
- \(\color{blue}{x^{2} + 10x + 25}\): first term \(\color{blue}{x^{2}}\) ✓, last term \(\color{blue}{25 = 5^{2}}\) ✓, middle term \(\color{blue}{2(x)(5) = 10x}\) ✓ → perfect square.
- \(\color{blue}{x^{2} – 14x + 49}\): last term \(\color{blue}{49 = 7^{2}}\), middle \(\color{blue}{2(x)(7) = 14x}\) ✓ → perfect square.
- \(\color{blue}{4x^{2} + 12x + 9}\): first \(\color{blue}{(2x)^{2}}\) ✓, last \(\color{blue}{3^{2}}\) ✓, middle \(\color{blue}{2(2x)(3) = 12x}\) ✓ → perfect square.
Factoring a Perfect Square Trinomial
Once recognized, factor directly using the pattern:
- \(\color{blue}{x^{2} + 10x + 25 = (x + 5)^{2}}\)
- \(\color{blue}{x^{2} – 14x + 49 = (x – 7)^{2}}\)
- \(\color{blue}{4x^{2} + 12x + 9 = (2x + 3)^{2}}\)
- \(\color{blue}{9x^{2} – 24x + 16 = (3x – 4)^{2}}\)
Step-by-Step Summary
- Identify \(\color{blue}{a}\) as the square root of the first term.
- Identify \(\color{blue}{b}\) as the square root of the last term.
- Verify the middle term equals \(\color{blue}{\pm 2\text{ ab }}\).
- If positive middle: \(\color{blue}{(a + b)^{2}}\). If negative middle: \(\color{blue}{(a – b)^{2}}\).
Watch: Factoring Perfect Square Trinomials (Video Lesson)
Mario’s Math Tutoring walks through several factoring examples at a clear, approachable pace:
Perfect Square Trinomial — Worked Examples
Example 1: Factor \(\color{blue}{x^{2} + 10x + 25}\).
\(\color{blue}{a = x}\), \(\color{blue}{b = 5}\). Check: \(\color{blue}{2(x)(5) = 10x}\) ✓.
Answer: \(\color{blue}{(x + 5)^{2}}\)
Example 2: Factor \(\color{blue}{x^{2} – 14x + 49}\).
\(\color{blue}{a = x}\), \(\color{blue}{b = 7}\). Check: \(\color{blue}{2(x)(7) = 14x}\) ✓. Middle term is negative.
Answer: \(\color{blue}{(x – 7)^{2}}\)
Example 3: Factor \(\color{blue}{4x^{2} + 12x + 9}\).
\(\color{blue}{a = 2x}\), \(\color{blue}{b = 3}\). Check: \(\color{blue}{2(2x)(3) = 12x}\) ✓.
Answer: \(\color{blue}{(2x + 3)^{2}}\)
Example 4: Expand \(\color{blue}{(3x – 4)^{2}}\).
Use \(\color{blue}{(a – b)^{2} = a^{2} – 2\text{ ab } + b^{2}}\): \(\color{blue}{(3x)^{2} – 2(3x)(4) + 4^{2} = 9x^{2} – 24x + 16}\).
More Examples: Polynomial Special Products Video
Khan Academy explores perfect square trinomials and other special products in this algebra lesson:
Exercises for Perfect Square Trinomials
Factor each perfect square trinomial or expand each perfect square binomial.
- \(\color{blue}{x^{2} + 8x + 16}\)
- \(\color{blue}{x^{2} – 12x + 36}\)
- \(\color{blue}{4x^{2} + 20x + 25}\)
- \(\color{blue}{9x^{2} – 6x + 1}\)
- \(\color{blue}{16x^{2} + 40x + 25}\)
Answers
- \(\color{blue}{(x + 4)^{2}}\)
- \(\color{blue}{(x – 6)^{2}}\)
- \(\color{blue}{(2x + 5)^{2}}\)
- \(\color{blue}{(3x – 1)^{2}}\)
- \(\color{blue}{(4x + 5)^{2}}\)
Free Perfect Square Trinomial Worksheet
Ready to practice on your own? Download our free Perfect Square Trinomial worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Perfect Square Trinomial before a quiz or test.
Download Special Products of Polynomials Worksheet
Frequently Asked Questions
What is a perfect square trinomial?
A perfect square trinomial is a three-term expression you get by squaring a binomial. It follows one of two patterns: a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². For example, x² + 6x + 9 = (x + 3)², because x² and 9 are squares and the middle term 6x equals 2 × x × 3.
How do you solve a perfect square trinomial step by step?
First, check that the first and last terms are perfect squares and the middle term is twice their product. Next, factor into a squared binomial. Then set it equal to zero and solve. For x² − 10x + 25 = 0: 25 = 5² and 10x = 2 × x × 5, so it factors as (x − 5)² = 0. That gives one repeated solution, x = 5.
How do you factor perfect square trinomials when a is not 1?
Take the square root of the first and last terms, then check the middle term. For 4x² + 12x + 9, the roots are 2x and 3. Twice their product is 2 × 2x × 3 = 12x, which matches, so 4x² + 12x + 9 = (2x + 3)². If the middle term does not match, the trinomial is not a perfect square and needs another factoring method.
How do you solve a perfect square trinomial with fractions?
The pattern works the same way with fractions. In x² + x + 1/4, the last term is (1/2)², and the middle term x equals 2 × x × 1/2, so it factors as (x + 1/2)². To solve x² + x + 1/4 = 0, set x + 1/2 = 0 and get x = −1/2. Clearing fractions first by multiplying by 4 also works: 4x² + 4x + 1 = (2x + 1)².
Can a perfect square trinomial be negative?
The middle term can be negative, as in x² − 8x + 16 = (x − 4)², but the first and last terms of a perfect square trinomial are always positive because they are squares. Its value is also never negative for real x, since a square is at least 0. An expression like −x² − 6x − 9 equals −(x + 3)², the negative of a perfect square trinomial.
How do you solve quadratics with perfect square trinomials?
Use completing the square to build one. For x² + 8x = 9, take half of 8, which is 4, and square it to get 16. Add 16 to both sides: x² + 8x + 16 = 25, so (x + 4)² = 25. Take square roots: x + 4 = ±5, giving x = 1 or x = −9. Both check in the original equation.
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