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.

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:

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• \(\color{blue}{(a + b)^{2} = a^{2} + 2\text{ ab } + b^{2}}\)
• \(\color{blue}{(a – b)^{2} = a^{2} – 2\text{ ab } + b^{2}}\)

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

1. The first term must be a perfect square: \(\color{blue}{a^{2}}\).
2. The last term must be a positive perfect square: \(\color{blue}{b^{2}}\).
3. 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

1. Identify \(\color{blue}{a}\) as the square root of the first term.
2. Identify \(\color{blue}{b}\) as the square root of the last term.
3. Verify the middle term equals \(\color{blue}{\pm 2\text{ ab }}\).
4. 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.

1. \(\color{blue}{x^{2} + 8x + 16}\)
2. \(\color{blue}{x^{2} – 12x + 36}\)
3. \(\color{blue}{4x^{2} + 20x + 25}\)
4. \(\color{blue}{9x^{2} – 6x + 1}\)
5. \(\color{blue}{16x^{2} + 40x + 25}\)

Answers

1. \(\color{blue}{(x + 4)^{2}}\)
2. \(\color{blue}{(x – 6)^{2}}\)
3. \(\color{blue}{(2x + 5)^{2}}\)
4. \(\color{blue}{(3x – 1)^{2}}\)
5. \(\color{blue}{(4x + 5)^{2}}\)

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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

How do I tell a perfect square trinomial from an ordinary trinomial?

Check whether the first and last terms are both perfect squares, and whether the middle term equals exactly ±2 times the product of their square roots. If all three conditions hold, it is a perfect square trinomial.

Can the last term of a perfect square trinomial be negative?

No. In \(\color{blue}{(a \pm b)^{2}}\), the last term is always \(\color{blue}{b^{2}}\), which is non-negative. A trinomial with a negative last term cannot be a perfect square trinomial.

What is the connection between perfect square trinomials and completing the square?

Completing the square deliberately creates a perfect square trinomial on one side of an equation so you can factor it instantly as \(\color{blue}{(x + k)^{2}}\) and then solve by taking square roots.

Related Topics

• Factoring Trinomials
• Factoring Polynomials
• Multiplying Binomials
• Simplifying Polynomials
• Adding and Subtracting Polynomials