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Finding Squares Using (a + d)(a - d) + d²

High School Student reading a maths blog

This interactive guide reveals a smart algebraic shortcut to find the squares of numbers up to 200 — without long multiplication. Select a number and watch each step unfold visually — you'll be surprised how quick and elegant this trick is!

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 and Squares and Cubes Reference to learn and memorize.

The Trick

This technique is based on a derived algebraic identity:

a2 = (a + d)(a - d) + d2

Where d is the difference between a and the nearest tens number.

This trick works from left to right.

Many mathemagicians and mental-math experts prefer solving problems from left to right, starting with the higher place values first. It feels more natural for quick, in-mind calculations and keeps the focus on the number's overall size — just as the brain processes numbers intuitively.

Choose a number

The steps are...

Let a be 64. Finding a2

64✕64

Take d = 4 and apply the algebraic identity:

a2 = (a + d)(a - d) + d2
(64 + 4) (64 - 4) + 42

This d can be any single digit number. We choose 4 so that we can have an easy multiplication with 60 in the next step.

68✕60+16
4080+16

And the solution is:

4096
642 = 4096

Solution for all the given numbers

InitializeApply the identitySolution
  • Finding 172
  • Let a = 17
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (17 + 7)(17 - 7) + 72
  • = (24)(10) + 49
  • = 240 + 49
289
  • Finding 282
  • Let a = 28
  • Let d = 8
  •   (a + d)(a - d) + d2
  • = (28 + 8)(28 - 8) + 82
  • = (36)(20) + 64
  • = 720 + 64
784
  • Finding 242
  • Let a = 24
  • Let d = 4
  •   (a + d)(a - d) + d2
  • = (24 + 4)(24 - 4) + 42
  • = (28)(20) + 16
  • = 560 + 16
576
  • Finding 342
  • Let a = 34
  • Let d = 4
  •   (a + d)(a - d) + d2
  • = (34 + 4)(34 - 4) + 42
  • = (38)(30) + 16
  • = 1140 + 16
1156
  • Finding 392
  • Let a = 39
  • Let d = 9
  •   (a + d)(a - d) + d2
  • = (39 + 9)(39 - 9) + 92
  • = (48)(30) + 81
  • = 1440 + 81
1521
  • Finding 472
  • Let a = 47
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (47 + 7)(47 - 7) + 72
  • = (54)(40) + 49
  • = 2160 + 49
2209
  • Finding 552
  • Let a = 55
  • Let d = 5
  •   (a + d)(a - d) + d2
  • = (55 + 5)(55 - 5) + 52
  • = (60)(50) + 25
  • = 3000 + 25
3025
  • Finding 642
  • Let a = 64
  • Let d = 4
  •   (a + d)(a - d) + d2
  • = (64 + 4)(64 - 4) + 42
  • = (68)(60) + 16
  • = 4080 + 16
4096
  • Finding 672
  • Let a = 67
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (67 + 7)(67 - 7) + 72
  • = (74)(60) + 49
  • = 4440 + 49
4489
  • Finding 712
  • Let a = 71
  • Let d = 1
  •   (a + d)(a - d) + d2
  • = (71 + 1)(71 - 1) + 12
  • = (72)(70) + 1
  • = 5040 + 1
5041
  • Finding 772
  • Let a = 77
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (77 + 7)(77 - 7) + 72
  • = (84)(70) + 49
  • = 5880 + 49
5929
  • Finding 832
  • Let a = 83
  • Let d = 3
  •   (a + d)(a - d) + d2
  • = (83 + 3)(83 - 3) + 32
  • = (86)(80) + 9
  • = 6880 + 9
6889
  • Finding 882
  • Let a = 88
  • Let d = 8
  •   (a + d)(a - d) + d2
  • = (88 + 8)(88 - 8) + 82
  • = (96)(80) + 64
  • = 7680 + 64
7744
  • Finding 922
  • Let a = 92
  • Let d = 2
  •   (a + d)(a - d) + d2
  • = (92 + 2)(92 - 2) + 22
  • = (94)(90) + 4
  • = 8460 + 4
8464
  • Finding 962
  • Let a = 96
  • Let d = 6
  •   (a + d)(a - d) + d2
  • = (96 + 6)(96 - 6) + 62
  • = (102)(90) + 36
  • = 9180 + 36
9216
  • Finding 1122
  • Let a = 112
  • Let d = 2
  •   (a + d)(a - d) + d2
  • = (112 + 2)(112 - 2) + 22
  • = (114)(110) + 4
  • = 12540 + 4
12544
  • Finding 1082
  • Let a = 108
  • Let d = 8
  •   (a + d)(a - d) + d2
  • = (108 + 8)(108 - 8) + 82
  • = (116)(100) + 64
  • = 11600 + 64
11664
  • Finding 1242
  • Let a = 124
  • Let d = 4
  •   (a + d)(a - d) + d2
  • = (124 + 4)(124 - 4) + 42
  • = (128)(120) + 16
  • = 15360 + 16
15376
  • Finding 1372
  • Let a = 137
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (137 + 7)(137 - 7) + 72
  • = (144)(130) + 49
  • = 18720 + 49
18769
  • Finding 1422
  • Let a = 142
  • Let d = 2
  •   (a + d)(a - d) + d2
  • = (142 + 2)(142 - 2) + 22
  • = (144)(140) + 4
  • = 20160 + 4
20164
  • Finding 1482
  • Let a = 148
  • Let d = 8
  •   (a + d)(a - d) + d2
  • = (148 + 8)(148 - 8) + 82
  • = (156)(140) + 64
  • = 21840 + 64
21904
  • Finding 1582
  • Let a = 158
  • Let d = 8
  •   (a + d)(a - d) + d2
  • = (158 + 8)(158 - 8) + 82
  • = (166)(150) + 64
  • = 24900 + 64
24964
  • Finding 1632
  • Let a = 163
  • Let d = 3
  •   (a + d)(a - d) + d2
  • = (163 + 3)(163 - 3) + 32
  • = (166)(160) + 9
  • = 26560 + 9
26569
  • Finding 1672
  • Let a = 167
  • Let d = 7
  •   (a + d)(a - d) + d2
  • = (167 + 7)(167 - 7) + 72
  • = (174)(160) + 49
  • = 27840 + 49
27889
  • Finding 1792
  • Let a = 179
  • Let d = 9
  •   (a + d)(a - d) + d2
  • = (179 + 9)(179 - 9) + 92
  • = (188)(170) + 81
  • = 31960 + 81
32041
  • Finding 1722
  • Let a = 172
  • Let d = 2
  •   (a + d)(a - d) + d2
  • = (172 + 2)(172 - 2) + 22
  • = (174)(170) + 4
  • = 29580 + 4
29584
  • Finding 1812
  • Let a = 181
  • Let d = 1
  •   (a + d)(a - d) + d2
  • = (181 + 1)(181 - 1) + 12
  • = (182)(180) + 1
  • = 32760 + 1
32761
  • Finding 1962
  • Let a = 196
  • Let d = 6
  •   (a + d)(a - d) + d2
  • = (196 + 6)(196 - 6) + 62
  • = (202)(190) + 36
  • = 38380 + 36
38416