Squaring Tricks

11 - 19

  1. Square the ones digit. This becomes your last digit.
  2. Multiply the ones digit by 2. This becomes your second digit.
  3. Retain the tens digit (i.e., 1) and append the above results.

Example: \(17^{2}\)

  1. \(7^{2}=49\). The 9 is your last digit. Carry the 4 to the left.
  2. \(7 \times 2 = 14 \). The 4 plus the 4 remaining from the first step is your second digit, 8. As before carry the 1 to the left.
  3. Retain the 1 from "17", add to it the 1 remaining from the previous step, and you get 289.

20 - 50

  1. Subtract the number you want to square from 100. (100 - x)
  2. Take this difference and subtract it from the number you want to square. [x - (100 - x)]. This is the first 2 digits of your answer.
  3. Square the difference, the answer from step 1. This is the last 2 digits of your answer.

Example 1: \(96^{2}\)

  1. \(100 - 96 = 4 \)
  2. \(96 - 4 = 92\). First two digits of answer.
  3. \( 4^{2} = 16 \). Last two digits of answer. \(96^{2} = 9216\)

Example 2: \(88^{2}\)

  1. \(100 - 88 = 12\)
  2. \(88 - 12 = 76\). First two digits of answer.
  3. \(12^{2} = 144\). 44 are the last two digits. Carry the 1 over the 6 in previous result to obtain 7744.

50 - 59

  1. Square the tens digit and add to that the ones digit if your original number. That's the first two numbers of your answer.
  2. Square your ones digit. That's your last two numbers.

Example: \(54^{2}\)

  1. \( 5^{2} + 4 = 29 \). That's our first two numbers.
  2. \( 4^{2} = 16 \). That's our last two numbers. The answer is 2916.

Numbers greater than 100

  1. Subtract 100 from the number you want to square (x - 100).
  2. Take the difference and add it to the number you want to square. [x + (x - 100)]. This is the first 3 digits of your answer.
  3. Square the difference, the answer from step 1. This is the last 2 digits of your answer.

Example: \(112^{2}\)

  1. \(112 - 100 = 12\)
  2. \(112 + 12 = 124 \). First three digits of answer.
  3. \( 12^{2} = 144 \). Carry the 1 over the 4. Answer: 12,544

Numbers around 1000

  1. Subtract 1000 from the number you want to square (x - 1000).
  2. Take the difference and add it to the number you want to square. [x + (x - 100)]. This is the first 4 digits of your answer.
  3. Square the difference, the answer from step 1. This is the last 3 digits of your answer.

Example 1: \(996^{2}\)

  1. \(1000 - 996 = 4 \)
  2. \(996 - 4 = 992\). First 3 digits.
  3. \(4^{2}= 16\). Last 3 digits. Answer: 992,016

Example 2: \(1007^{2}\)

  1. \(1007 - 1000 = 7 \)
  2. \(1007 + 7 = 1014\). First 4 digits.
  3. \(7^{2} = 49\). Last 3 digits. Answer: 1,014,049

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