(a + b)² Formula: Expansion, Proof & Solved Examples

The (a + b)² formula says that (a + b)² = a² + 2ab + b². In words: the square of a sum equals the square of the first term, plus twice the product of the two terms, plus the square of the second term.

This one identity appears from Class 8 all the way to Class 12, so it is worth understanding properly rather than just memorising.

What Does (a + b)² Mean?

(a + b)² simply means (a + b) multiplied by itself: (a + b)² = (a + b) × (a + b)

Many students write (a + b)² = a² + b². That is the single most common algebra mistake in school. The middle term 2ab is missing, and it changes the answer completely.

Quick check with numbers: take a = 2 and b = 3.

• (2 + 3)² = 5² = 25
• 2² + 3² = 4 + 9 = 13 ✗
• 2² + 2(2)(3) + 3² = 4 + 12 + 9 = 25 ✓

Proof 1: Algebraic Proof (Multiplying It Out)

Multiply each term of the first bracket by each term of the second:

(a + b)(a + b) = a × a + a × b + b × a + b × b = a² + ab + ab + b² = a² + 2ab + b²

The two "ab" terms combine, and that is where the 2ab comes from.

Proof 2: The Geometric (Square-Area) Proof

This picture proof is the one that makes children say "Oh, now I get it!"

1. Draw a big square with each side of length (a + b). Its area is (a + b)².
2. Mark a point on each side so the side splits into a part of length a and a part of length b.
3. Join the points with lines. The big square now breaks into four pieces: one square of side a (area a²), one square of side b (area b²), and two rectangles of size a × b (area ab each).
4. Add the four areas: a² + b² + ab + ab = a² + 2ab + b²

Both proofs give the same answer, so (a + b)² = a² + 2ab + b².

Tip for parents: cut this out of paper with your child. Use one colour for a², another for b² and a third for the two rectangles. The formula sticks far better after a 10-minute activity than after 10 repetitions.

6 Solved Examples

Example 1: Expand (x + 5)²
Here a = x and b = 5.
(x + 5)² = x² + 2(x)(5) + 5² = x² + 10x + 25

Example 2: Expand (2a + 3b)²
(2a + 3b)² = (2a)² + 2(2a)(3b) + (3b)² = 4a² + 12ab + 9b²
Watch out: (2a)² is 4a², not 2a².

Example 3: Find 103² in your head
Break 103 into 100 + 3.
103² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609

Example 4: Find 51² mentally
51 = 50 + 1
51² = 2500 + 2(50)(1) + 1 = 2500 + 100 + 1 = 2601

Example 5: Find 98² using (a − b)²
98 = 100 − 2, so use (a − b)² = a² − 2ab + b².
98² = 10000 − 400 + 4 = 9604

Example 6: If a + b = 7 and ab = 10, find a² + b²
Rearrange: a² + b² = (a + b)² − 2ab = 49 − 20 = 29
Check: 5 and 2 add to 7 and multiply to 10, and 5² + 2² = 25 + 4 = 29 ✓

Mental Maths Tricks Using (a + b)²

Number	Split it as	Working	Answer
21²	20 + 1	400 + 40 + 1	441
32²	30 + 2	900 + 120 + 4	1024
105²	100 + 5	10000 + 1000 + 25	11025
1002²	1000 + 2	1000000 + 4000 + 4	1004004
99²	100 − 1	10000 − 200 + 1	9801

The trick works best when a number sits close to a "round" number like 10, 50, 100 or 1000.

Related Formulas You Should Know

Identity	Formula
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
a² + b²	(a + b)² − 2ab
a² + b²	(a − b)² + 2ab
(a + b)² − (a − b)²	4ab
(a + b)² + (a − b)²	2(a² + b²)
(a + b)(a − b)	a² − b²

For the full list by class, see our algebraic identities list for Class 8, 9 & 10. For the cube versions, read (a + b)³ and (a − b)³ formulas.

Practice Questions

1. Expand (y + 4)²
2. Expand (3x + 2)²
3. Find 41² mentally
4. If a + b = 10 and ab = 21, find a² + b²
5. Expand (p − 6)²

Answers: 1. y² + 8y + 16   2. 9x² + 12x + 4   3. 1681   4. 58   5. p² − 12p + 36