Splitting the Square
Find an equivalent expression for 3 6 2 .
Note: While you can do this on a calculator, try to work it out without one! This can generalize into a method for squaring numbers mentally.
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3 solutions
If A number then let Y represents the unit place digit and X represents the remaining digits then A 2 = ( 1 0 X + Y ) 2 = ( 1 0 X ) 2 + 2 0 X Y + ( Y ) 2 A 2 = 1 0 X ( 1 0 X + 2 Y ) + Y 2 = 1 0 X ( A + Y ) + Y 2 Thus, 3 6 2 = 3 0 × ( 3 6 + 6 ) + 6 2 = ( 3 0 × 4 2 ) + ( 6 × 6 )
The idea is to change the number to be squared, call it n , into a near multiple of 1 0 through either addition or subtraction, and then make use of the difference of squares pattern to re-write n 2 = ( n + α ) ( n − α ) + α 2 . In the given example, we can either use 3 6 + 4 = 4 0 or 3 6 − 6 = 3 0 . Then either
x 2 = ( x + 4 ) ( x − 4 ) + 4 2 ⟹ 3 6 2 = ( 3 6 + 4 ) ( 3 6 − 4 ) + 1 6 = ( 4 0 × 3 2 ) + 1 6 = 1 2 8 0 + 1 6 = 1 2 9 6
or
x 2 = ( x − 6 ) ( x + 6 ) + 6 2 ⟹ 3 6 2 = ( 3 6 − 6 ) ( 3 6 + 6 ) + 3 6 = ( 3 0 × 4 2 ) + 3 6 = 1 2 6 0 + 3 6 = 1 2 9 6
The latter is the calculation given in the answer choices; I might have used the former but it makes little difference.