Q: x^2 - 36^2 = 85*157

Answer 1

Q: x^2 - 36^2 = 85*157

Often times a math problem has a short cut. Perhaps this is one reason that many people consider one good at math smart. You can become smart this way by seeing and trying more such math problems.

You may skip below but directly go to the “=======“ part to check one short-cut approach to solve this problem. You are encouraged to join and post your other ideas. When you truly grasp this problem, you may design a similar problem and post it here. You can even try the problem that you designed with your friends, for fun!

On the surface, this problem seems challenging in the calculation, such big numbers, a lot of calculations, and we even don’t know whether we can find a calculator. The problem itself though is fairly straightforward. We simply need to move all the numbers onto one side of the equation, a move results in variables on the other side, as below,

      x^2 = 85*157 + 36^2

You see, this is just a problem of x^2 = some number. We can just manually calculate the right side and get 14641. Now the problem becomes to solve this below equation:

      x^2 = 14641

We know how we can easily get an answer if the equation is like:

     x^2 = 9      or     x^2 = 25

With the big number 14641, we do the similar. We ask ourselves, is this 14641 a square of some whole number. We wish it is, and even without a calculator we can still move forward from here and try it, by factoring this big number 14641. 

Here we go. We can start by trying, 1, 2, 3, ….. Wow, we got it when we get to 11. Therefore 14641 = 11*1331. This is not too bad, giving us a hope that 1331 might be able to be further factorized. So we now do the similar to factor 1331, by trying, 1, 2, 3, …. Wow, we got another 11! 1331 = 11*121.

Now perhaps you see 121 = 11*11 if you already have memorized the table of squares. Or you can just repeat what we did above to factor 121 by trying, 1, 2, 3, ….. and get 11*11.

Therefore, we now simplified the equation 

from    x^2 = 14641   to    x^2 = 11*11*11*11 = (11*11)^2

We now get x = 121, or -121. 

    

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Instead of the calculation we used in above approach, here is a short cut if you are familiar with the formula of squares of binomial. 

The original equation can be transferrred along below steps:

             x^2 - 36^2 = 85 * 157

x^2 - 36^2 = 85 * 157 = 85*(85 + 72) = 85*(85+2*36) = 85^2 + 2*85*36

Hence,  now moving 36^2 to the other side of the equation we have 

x^2 = 85^2 + 2*85*36 + 36^2 = (85+36)^2 = 121^2

x = 121, or -121

How do we know this short cut? Honestly, initially we don’t know for sure. We got here by 1) trying, and 2) trying on a reasonable direction. 

How do we get on this reasonable direction? From our familiarity with and solid understanding on the formula of squares of binomial. 

You see, math has no trick. Everyone can get it, starting from the very basic math rules. Try it yourself to design a similar math problem to have fun with your friends! 

BONUS: Here the foundation of this problem is (a+b)^2 = a^2 + 2ab + b^2. How about design a similar math problem using the foundation of (a-b)^2 = a^2 -2ab + b^2?

BIG BONUS: Here is an even easier way to solve this problem, if you are familiar with the formula called difference of squares: a^2 - b^2 = (a-b)*(a+b). Give it a try!

Enjoy math :)