# Squaring a Trinomial: Formula and Examples

TL;DR

Squaring a trinomial applies the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca, three squares plus twice each pair-product. This article derives the formula, works through examples with numbers and variables, and shows why the cross terms are the part everyone forgets.

## What Is Squaring a Trinomial?

A **trinomial** is a polynomial with exactly three terms, for example x+y+z or x² + 2x + 3. **Squaring** it means raising the whole expression to the power 2, so (a+b+c)² is shorthand for (a+b+c)(a+b+c).

The result is governed by one of the standard [algebraic identities](/content/math/algebra/algebraic-identities/index.html): (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Two things are worth naming right away. The **squared terms** a², b², c² come from each term meeting itself. The **cross terms** 2ab, 2bc, 2ca come from each pair of different terms, and the factor of 2 appears because each pair shows up twice in the full multiplication.

## How Do You Derive the Squaring a Trinomial Formula?

Treat (a+b+c)(a+b+c)(a+b+c) as (a+(b+c)) and use the binomial square: (a+(b+c))² = a² + 2a(b+c) + (b+c)².

Expand the last piece with (b+c)² = b² + 2bc + c²:
a² + 2ab + 2ac + b² + 2bc + c².

Reorder into squares first, then pairs: a² + b² + c² + 2ab + 2bc + 2ca.

That is the identity. Every squared term appears once; every cross term appears with a 2. This is the same [multiplying-polynomials](/content/math/algebra/multiplying-polynomials/index.html) skill applied to a three-term factor.

## Examples of Squaring a Trinomial

### Example 1

**Expand (x+y+z)².**  
Square each term: x², y², z².  
Twice each pair: 2xy, 2yz, 2zx.

(x+y+z)² = x² + y² + z² + 2xy + 2yz + 2zx.  
**Final answer:** as above.

### Example 2

**A student expands (2+3+4)² as 2² + 3² + 4² = 29.**  
The tempting move is to square each term and add, it feels complete.

That gives 4 + 9 + 16 = 29.  
Check it against the direct value: 2 + 3 + 4 = 9, and 9² = 81, not 29.

The gap is the cross terms. Add them: 2(2⋅3) + 2(3⋅4) + 2(4⋅2) = 12 + 24 + 16 = 52.

Now 29 + 52 = 81.  
**Final answer:** 81. Squaring the terms and stopping drops the six cross-terms, and the answer is off by more than half.

### Example 3

**Expand (x + 2y + 3)².**  
Set a = x, b = 2y, c = 3.

Squares: x², (2y)² = 4y², 3² = 9.
Pairs: 2(x)(2y) = 4xy, 2(2y)(3) = 12y, 2(3)(x) = 6x.

(x + 2y + 3)² = x² + 4y² + 9 + 4xy + 12y + 6x.  
**Final answer:** as above.

### Example 4

**Expand (a+b−c)².**  
A subtracted term is a negative term, so set c→−c.

Squares: a², b², (−c)² = c².
Pairs: 2ab, 2(b)(−c) = −2bc, 2(−c)(a) = −2ca.

(a+b−c)² = a² + b² + c² + 2ab − 2bc − 2ca.  
**Final answer:** the cross terms involving c flip sign; the squares stay positive.

### Example 5

**Expand (x² + 2x + 3)².**  
Set a = x², b = 2x, c = 3.

Squares: x⁴, 4x², 9.
Pairs: 2(x²)(2x) = 4x³, 2(2x)(3) = 12x, 2(3)(x²) = 6x².
Combine the x² terms: 4x² + 6x² = 10x².

(x² + 2x + 3)² = x⁴ + 4x³ + 10x² + 12x + 9.  
**Final answer:** as above.

### Example 6

**A square field has side (p+q+5) metres. Write its area.**  
Area of a square is side squared.

(p+q+5)² = p² + q² + 25 + 2pq + 10q + 10p.  
**Final answer:** p² + q² + 25 + 2pq + 10p + 10q square metres.

## Why the Cross Terms Are the Whole Point

The formula matters because the cross terms carry real quantity, they are not decoration.

- **In geometry, they are area.** Split a square of side (a+b+c) into a grid, and the cross terms are the rectangular strips between the corner squares. Leave them out and you have lost pieces of the square.
- **In algebra, they connect to the** [**perfect square trinomial**](/content/math/algebra/perfect-square-trinomial/index.html). The binomial case is the same idea with one cross term; the trinomial has three.
- **In error analysis, dropped cross terms compound.** When engineers linearise a formula and discard the "small" cross terms too early, the leftover error grows, as seen in the [Tacoma Narrows Bridge](https://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)) collapse of 1940.

## Where the Expansion Goes Wrong

Every common error traces back to the cross terms.

### Mistake 1: Distributing the square over the sum

**Where it slips in:** the very first step.

**Don't do this:** write (a+b+c)²=a²+b²+c².  
**The correct way:** a square never distributes over addition.

### Mistake 2: Missing a pair

**Where it slips in:** listing the cross terms from memory.

**Don't do this:** write only 2ab + 2bc and forget 2ca.

### Mistake 3: Mishandling a negative term

**Where it slips in:** expanding (a−b+c)².

**Don't do this:** treat −b as b and copy the all-positive formula.  
**The correct way:** carry the sign into the term.

## Conclusion
- **Squaring a trinomial** uses the identity (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca.
- The three squares are straightforward; the three doubled cross-terms are frequently forgotten.
- A negative term keeps its square positive but flips the sign of its cross-products.