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(a + b)² Formula — a²+2ab+b², Proof, Examples

What Is the (a + b)² Formula?

The a plus b whole square formula expands the square of a binomial (a+b) into a trinomial. For any real (or even complex) numbers a and b:

(a+b)² = a² + 2ab + b².

Each term has a name worth knowing:

• a² — the first square term, the square of the first quantity.
• 2ab — the middle term (or cross term), twice the product of the two quantities.
• b² — the second square term, the square of the second quantity.

The companion identity is (a−b)² = a²−2ab+b² — same shape, with the middle term flipped to minus.

How Is the (a + b)² Formula Derived?

Two proofs are worth seeing — one algebraic, one geometric. They explain the same fact from different directions, and together they make the 2ab impossible to forget.

Algebraic proof — just multiply it out. Expand using the distributive property:

(a+b)(a+b) = a⋅a + a⋅b + b⋅a + b⋅b = a² + ab + ba + b².

Since ab = ba, the two middle terms combine:

=a² + 2ab + b².

Geometric proof — the area of a square. Draw a square whose side is (a+b). Its total area is (a+b)². Now cut it with one horizontal and one vertical line at the point that splits each side into a length a and a length b. The square falls into four pieces:

• one a×a square, area a²,
• one b×b square, area b²,
• two a×b rectangles, area ab each, totaling 2ab.

The pieces must add up to the whole:

(a+b)² = a² + 2ab + b².

How Is (a + b)² Different from a² + b²?

This is the distinction the whole topic turns on. They are not equal:

(a+b)² = a² + 2ab + b²,

a² + b² = (a+b)² − 2ab.

The gap between them is exactly 2ab. A quick numeric check settles it: with a=3, b=4, (3+4)² = 49 but 3² + 4² = 25.

Examples of the (a + b)² Formula

Example 1

Expand (x+5)². Apply the formula with a=x, b=5:

(x+5)² = x² + 10x + 25.

Final answer: x² + 10x + 25.

Example 2

Expand (3x+4)². Use the formula with a=3x, b=4:

(3x+4)² = 9x² + 24x + 16.

Final answer: 9x² + 24x + 16.

Example 3

Use the formula to compute 52² mentally. Write 52=50+2 and apply the identity with a=50, b=2:

52² = 2704.

Final answer: 52² = 2704.

Example 4

Expand (2x+3y)². With a=2x, b=3y:

(2x+3y)² = 4x² + 12xy + 9y².

Final answer: 4x² + 12xy + 9y².

Example 5

Given a+b=7 and ab=10, find a²+b². Rearrange the identity:

a²+b² = 29.

Final answer: a² + b² = 29.

Example 6

Factor x²+14x+49 back into a whole square. Recognise the pattern:

x² + 14x + 49 = (x + 7)².

Final answer: (x + 7)².

Where the (a + b)² Formula Shows Up

• Completing the square.
• Mental arithmetic.
• Statistics — variance.
• Geometry and physics.

Tripping Points to Avoid

Mistake 1: Dropping the middle term

Don't do this: Writing (a+b)² = a² + b².

The correct way: Always write all three terms, a² + 2ab + b².

Mistake 2: Confusing (a + b)² with (a − b)²

The correct way: (a+b)² = a² + 2ab + b²; (a−b)² = a² − 2ab + b².

Mistake 3: Misreading the cross term when terms have coefficients

The correct way: Treat the whole term as a.

Conclusion

• The a plus b whole square formula is (a+b)² = a² + 2ab + b² — two squares plus twice the product.
• The 2ab middle term comes from the two cross-products.
• The most common errors are dropping the middle term, confusing it with (a−b)², and losing the variable in the cross term.

Practice These Before Moving On

1. Expand (2x+7)².
2. Compute 10³.
3. Given a+b=9 and ab=14, find a²+b².