Using the Difference of Squares Factoring Pattern to Simplify a Mental Math Calculation

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Today's suspiciously nice mental math calculation is an example of a factoring pattern called the difference of two squares (or sometimes just the difference of squares).

This pattern, or identity, is a² – b² = (a – b)(a + b). In other words, if you have a difference of two perfect squares, it will be equivalent to the sum of the square roots times the difference of the square roots. This often comes in handy for algebraic simplifications, but can also be used, as here, for mental math calculations.

#mentalmath #numbersense #differenceofsquares

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[Robair]

(a^2-b^2)=(a+b) (a-b) This is the formula he used

[AdamJohn-ieue]

"I know I'm a huge hit at parties" got me dying

[younickbongo]

Double and half trick 👍👍👍 love the name

[DhairyaPandya]

The double and half trick is impressive, am a math student last year high school but till the day never saw anyone use that lol but works!

[elderwand]

Thanks! Your double and half method is unique, you can starta series of similar vedic math tricks, might be helpful

[spelunkerd]

I like visualization of how one calculates the area of a square box measuring 39 on one side, minus a small square at one corner measuring 11 on the side. You get an L shaped area that is calculated by adding two rectangles together. The dimension of those rectangles added together is the "difference of the two squares". Ergo, the geometric origin of the term.

[joeschmo]

Yeh, I do the double/half trick all the time, usually for back-of-napkin approximations. So if one number is 13, doubling it is 26 (hmm, close to 25, so DH to ~50, DH again to ~100, and whatever the other number is, just multiply by 100 after halving/halving/halving, and that's a close-enough approximation.

[tabaahigaming]

I use the double and half trick every time to calculate stuff it's my favourite trick I am using it since class 7

[alexungureanu]

I learned this metod in the last months in school

[ahmedseen]

Aww I’ve used that double and half trick for so long, and still use it all the time :)
There’s a lot of fun with two’s and fives I’ve found. When multiplying by 25 you could do the same trick, quadruple and quarter. Because 25 is 5 squared, and 4 is 2 squared.
68x25 = 17x100 = 1700
When dividing by 5, you could multiply the numerator by 2, and set it back one decimal point, to get your answer.
141/5 = 28.2
So when you wanna divide by 25 you can multiply by 4, and set it back two decimal points.
602/25 = 24.08
When you divide by 125, you multiply by 8, set it back three decimal points.
67/125 = 0.536
And this could go on forever if you wanted it to, but obviously 2, 4 and 8 are the easiest powers of two to multiply.
It’s because these numbers are powers of 2 and 5, and those numbers have a special relationship in decimal, AKA base ten.

[bertrandspuzzle]

I love the shock I get when first applying this to Pythagorean's Theorem. A few kids the "oh this could be useful" reaction.

[JAdamMoore]

❤ x^2 - y^2 = p^n where x=(1/2 (p+1) p^((-1+n)/2)), y=(1/2 (p-1) p^((-1+n)/2))
This gives the pattern for n=3 as

3^2 - 1^2 = 2^3
6^2 - 3^2 = 3^3
10^2 - 6^2 = 4^3
15^2 - 10^2 = 5^3
...
resulting in the general formula described above for any p^n 😊

[PfSype]

Finally i saw new concept 😊, my dumbass was using calculus

[artemetra]

yay, i am not the only one who's using the double/half trick :') discoveted it on my own a couple of years ago and kept using it

[ztheyoutuber]

I used the double half method for a long time probably since I was in 4th grade

[chrissystewart]

Hello sir 😊 I'm new & I never learn this in school this is my first time 😊 I hope to passed my GED. I need to work on my math more

[TheEulerID]

In this case I just say 39^2 - 11^2 is (40 -1)^2 - 121 (as I learnt my 12 times table, not all the squares above 12).

(40-1)^2 - 121 is just 1600 - 80 + 1 - 121 = 1400.

As for the solution shown. Well, that's a very neat trick based on the difference of two squares, but it doesn't tell you anything fundamental about why you get a multiple of 100. That's just a property of the two numbers shown. if we chose 39^2 - 10^2 then the difference of two squares gives us :-

(39 - 10) x (39 + 10) = 29 x 49 = 29 x 50 - 29 = 1450 - 29 = 1421. Not a nice round number at all. The best I can say is that the answer will never be a prime number as the difference of two squares always gives you two factors.

[christopherrivera]

Can you do the same thing if it was a + sign

[charlessabinjr]

My sister just tells me that I'm weird, and then I smile and agree. 🤠

[Junjun]

How did I go from theoretical math to this 😅