Using the difference of two
squares, evaluate 21 multiplied by 19 without a calculator.
The difference of two squares
result or identity states that 𝑥 plus 𝑦 multiplied by 𝑥 minus 𝑦 is equal to 𝑥
squared minus 𝑦 squared. In this question, we’re trying to
multiply two numbers, 21 and 19. We can begin by letting 𝑥 plus 𝑦
equal 21 and 𝑥 minus 𝑦 equal 19. We have a pair of simultaneous
equations. And we can eliminate 𝑦 by
adding. This gives us two 𝑥 is equal to
40. Dividing both sides of this
equation by two gives us 𝑥 equals 20. Substituting this back into the
first equation gives us 20 plus 𝑦 is equal to 21. Subtracting 20 from both sides of
this equation gives us 𝑦 equals one.
The calculation 21 multiplied by 19
can be rewritten as 20 plus one multiplied by 20 minus one. Using the difference of two squares
result, this is equal to 20 squared minus one squared. 20 squared is equal to 400, and one
squared equals one. As 400 minus one is equal to 399,
21 multiplied by 19 must also be equal to 399.