# Perfect Square Trinomial — Formula & Examples

## TL;DR
A perfect square trinomial is a three-term expression that factors into the square of a binomial — a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a - b)². This article shows you the formula, how to recognize one on sight, how to factor it, and six worked examples.

## What Is a Perfect Square Trinomial?
A **perfect square trinomial** is a trinomial (a three-term polynomial) that equals the square of a binomial. In plain terms, it is what you get when you multiply a binomial like (a+b) by itself: (a + b)² = a² + 2ab + b². Two examples make the idea concrete:

1. The trinomial x² + 6x + 9 is a perfect square because it equals (x + 3)².
2. The trinomial x² − 10x + 25 is a perfect square because it equals (x − 5)².

## What Is the Perfect Square Trinomial Formula?
There are two forms, separated only by the sign of the middle term. Both come directly from squaring a binomial.

**Positive middle term** — from squaring a sum:
 
(a + b)² = a² + 2ab + b².

**Negative middle term** — from squaring a difference:

(a - b)² = a² - 2ab + b².

Here a is the square root of the first term, b is the square root of the last term, and 2ab is the middle term.

## How Do You Know If A Trinomial Is A Perfect Square?
Run three quick checks:

1. **First**: Is the leading term a perfect square?
2. **Second**: Is the constant term a perfect square?
3. **Third**: Does the middle term equal 2ab, twice the product of those two roots?

If all three hold, the trinomial is a perfect square.

## How Do You Factor a Perfect Square Trinomial?
Factoring is the formula run in reverse. Once a trinomial passes the test above, the factored form writes itself:

1. Take the square root of the first term.
2. Take the square root of the last term.
3. Read the sign of the middle term.
4. Write the binomial squared.

## Examples of Perfect Square Trinomial
### Example 1
**Factor x² + 8x + 16.**

Final answer: (x + 4)².

### Example 2
**Factor x² − 12x + 36.**

Final answer: (x − 6)².

### Example 3
**Factor 9x² − 6x + 1.**

Final answer: (3x − 1)².

### Example 4
**Factor 16a² − 40ab + 25b².**

Final answer: (4a − 5b)².

### Example 5
**Find the value of c that makes x² + 10x + c a perfect square trinomial.**

Final answer: c = 25.

### Example 6
**Is 4x² + 8x + 9 a perfect square trinomial?**

Final answer: No.

## Key Takeaways
- A **perfect square trinomial** factors into a binomial squared: a² ± 2ab + b² = (a ± b)².
- Identify it by checking: first term is a square, last term is a square, and middle term equals 2ab.
- Factor by taking the roots of the first and last terms.

## Practice These Before Moving On
1. Factor x² + 20x + 100.
2. Find the value of k that makes x² − 14x + k a perfect square trinomial.
3. Decide whether 25x² + 30x + 9 is a perfect square trinomial, and factor it if it is.