Squaring by shortcut

Squaring

 

Let us take advantage of  algebraic identity.

A² = (A - d)(A + d) + d²

Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply.

Examples. To square the number 23, we let d = 3 to get

23² = 20 x 26 + 3² = 520 + 9 = 529:

 

To square 48, let d = 2 to get

48² = 50 x 46 + 2² = 2300 + 4 = 2304:

 With just a little practice, it's possible to square any two-digit number in a matter of seconds. Once you have mastered those, we can quickly square three-digit numbers by rounding up and down to the nearest hundred.

Examples. 223² = 200 x 246 + 23²= 49;200 + 529 = 49;729

952² = 1000 x 904 + 48² = 904;000 + 2;304 = 906;304:

To do mental calculations of this size, one needs to be quick at multiplying 2-digit and 3-digit numbers by 1-digit numbers, generating the answer from left to right.