Watch with Theo · Shortcuts
ACT Math Shortcuts: The Difference of Squares
Use the difference of squares to turn a long multiplication into two simple steps.
Take this into your next practice session
Use a shortcut when you understand why it works. A direct calculation is always an option.
Choose an answer, then check your reasoning. This public practice is not saved.
Read all answer explanations
899 — correct answer: Correct. (30 − 1)(30 + 1) = 30² − 1² = 900 − 1 = 899.
900: The factors are one below and one above 30. Square 30, then subtract 1².
901: The factors are one below and one above 30. Square 30, then subtract 1².
841: The factors are one below and one above 30. Square 30, then subtract 1².
Full transcript and worked example
Look before you calculate.
Hi, I'm Theo. Before you reach for a calculator, look at the numbers. Forty nine times fifty one looks like a multiplication problem. It is also a pattern.
49 × 51
Find the number in the middle.
Forty nine is fifty minus one. Fifty one is fifty plus one. That makes these factors a matched pair, equally spaced around fifty.
(50 − 1)(50 + 1)
See why the shortcut works.
Multiply the brackets. You get a squared, plus a b, minus a b, minus b squared. The middle terms cancel. This is called the difference of squares.
(a − b)(a + b) = a² − b²
Square, then subtract.
Here a is fifty and b is one. Fifty squared is twenty five hundred. Subtract one squared, and the answer is twenty four ninety nine.
50² − 1² = 2500 − 1 = 2499
Use patterns you understand.
A shortcut is useful when you recognize why it works. Do not spend more time hunting for a trick than solving the problem. Now try twenty nine times thirty one. What number is in the middle?
29 × 31 = 30² − 1²
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