Finding Squares Using (a + b)²

This mathematics trick helps you calculate the squares of the numbers in your mind using just digits and carries — no calculator needed! Once it is learnt, you can easily write down the squares of the numbers up to 300.

What you need to know before you start...

You need to master Multiplication Tables (1 to 20) and Squares (1 to 30). Memorizing these essential concepts is crucial for any mathematics student. Not only will it benefit this blog post, but it will also prove invaluable for numerous other topics covered on our blogs.

Our specially crafted worksheets on Multiplication and Squares will help you solidify this knowledge.

Use the Times Tables Reference to learn and memorize.

The Trick

This 3-step technique is based on the classic algebraic identity:

(a + b)² = a² + 2ab + b²

But instead of calculating the whole expression at once, we split the number into digits and apply three steps.

The 3 steps are...

Finding 64²Break the given number 64 into 2 parts a and b: First part (up to tens place): 6 = aSecond part (units place): 4 = b

Step 1: b²
Square the second part b(4)
4² = 16, write 6 here and carry 1 to next step
Partial answer is ???6
Step 2: 2ab
Multiply the two parts by 2 and add carry from previous step2 ✕ 6 ✕ 4 = 48
48 + 1 = 49, write 9 here and carry 4 to next step
Partial answer is ??96
Step 3: a²
Square the first part a(6) and add carry from previous step
6² + 4 = 36 + 4 = 40, write 40 here
Final answer is 4096

64² = 4096

Solution for all the given numbers

Step	Finding 19²	Carry Over	Answer
1	9² = 81	8	??1
2	(2 ✕ 1 ✕ 9) + 8 = 26	2	?61
3	1² + 2 = 3		361

Step	Finding 28²	Carry Over	Answer
1	8² = 64	6	??4
2	(2 ✕ 2 ✕ 8) + 6 = 38	3	?84
3	2² + 3 = 7		784

Step	Finding 39²	Carry Over	Answer
1	9² = 81	8	???1
2	(2 ✕ 3 ✕ 9) + 8 = 62	6	??21
3	3² + 6 = 15		1521

Step	Finding 47²	Carry Over	Answer
1	7² = 49	4	???9
2	(2 ✕ 4 ✕ 7) + 4 = 60	6	??09
3	4² + 6 = 22		2209

Step	Finding 55²	Carry Over	Answer
1	5² = 25	2	???5
2	(2 ✕ 5 ✕ 5) + 2 = 52	5	??25
3	5² + 5 = 30		3025

Step	Finding 64²	Carry Over	Answer
1	4² = 16	1	???6
2	(2 ✕ 6 ✕ 4) + 1 = 49	4	??96
3	6² + 4 = 40		4096

Step	Finding 69²	Carry Over	Answer
1	9² = 81	8	???1
2	(2 ✕ 6 ✕ 9) + 8 = 116	11	??61
3	6² + 11 = 47		4761

Step	Finding 71²	Carry Over	Answer
1	1² = 1	0	???1
2	(2 ✕ 7 ✕ 1) + 0 = 14	1	??41
3	7² + 1 = 50		5041

Step	Finding 83²	Carry Over	Answer
1	3² = 9	0	???9
2	(2 ✕ 8 ✕ 3) + 0 = 48	4	??89
3	8² + 4 = 68		6889

Step	Finding 92²	Carry Over	Answer
1	2² = 4	0	???4
2	(2 ✕ 9 ✕ 2) + 0 = 36	3	??64
3	9² + 3 = 84		8464

Step	Finding 108²	Carry Over	Answer
1	8² = 64	6	????4
2	(2 ✕ 10 ✕ 8) + 6 = 166	16	???64
3	10² + 16 = 116		11664

Step	Finding 112²	Carry Over	Answer
1	2² = 4	0	????4
2	(2 ✕ 11 ✕ 2) + 0 = 44	4	???44
3	11² + 4 = 125		12544

Step	Finding 136²	Carry Over	Answer
1	6² = 36	3	????6
2	(2 ✕ 13 ✕ 6) + 3 = 159	15	???96
3	13² + 15 = 184		18496

Step	Finding 148²	Carry Over	Answer
1	8² = 64	6	????4
2	(2 ✕ 14 ✕ 8) + 6 = 230	23	???04
3	14² + 23 = 219		21904

Step	Finding 152²	Carry Over	Answer
1	2² = 4	0	????4
2	(2 ✕ 15 ✕ 2) + 0 = 60	6	???04
3	15² + 6 = 231		23104

Step	Finding 163²	Carry Over	Answer
1	3² = 9	0	????9
2	(2 ✕ 16 ✕ 3) + 0 = 96	9	???69
3	16² + 9 = 265		26569

Step	Finding 224²	Carry Over	Answer
1	4² = 16	1	????6
2	(2 ✕ 22 ✕ 4) + 1 = 177	17	???76
3	22² + 17 = 501		50176

Step	Finding 258²	Carry Over	Answer
1	8² = 64	6	????4
2	(2 ✕ 25 ✕ 8) + 6 = 406	40	???64
3	25² + 40 = 665		66564

Step	Finding 248²	Carry Over	Answer
1	8² = 64	6	????4
2	(2 ✕ 24 ✕ 8) + 6 = 390	39	???04
3	24² + 39 = 615		61504

Step	Finding 275²	Carry Over	Answer
1	5² = 25	2	????5
2	(2 ✕ 27 ✕ 5) + 2 = 272	27	???25
3	27² + 27 = 756		75625

Introduction

What you need to know before you start...

The Trick

Choose a number

The 3 steps are...

Solution for all the given numbers