(a+b)^2=a^2+2ab+b^2 But why?

So Lets prove it :
Consider a line 

Take any arbitrary point 

So now we can say that length of line is a+b
Now we have to square a+b
that will be

lets add all the squares and rectangles

This will be  a² 

Now it is a²+b²

Here there are two rectangle in the above diagram

As we know that Area of rectangle is Length X Width

So it will be a x b

Now lets add every square and rectangle

a² + b² + a x b + a x bi.e., a² + b² + 2ab

Thus 

(a+b)²=a² + 2ab+b² 

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Square of Number in 100's

Choose a number greater than 100.

For e.g.,

Lets start with 103

so

Step 1: Square the last two digits 03 X 03 = 09

Step 2: Add the last two number : 103 + 03 = 106

Step 3: Place the number in this format Step2 result Step1 result =10609 is our answer




Take Second example 112

so

Step 1: Square the last two digits 12 X 12 = 144
   
             Note : Only consider the last two digit of Step 1
                        and then the 100 ^th   place will be the carry.

Step 2: Add the last two number 112 + 12 = 124  + 1(carry) =125

Step 3: Place the number in the format Step 2 result (125) Step 1 result (44) =12544 is our answer....

Note: This may needs lots of practice to do further calculations.

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Squaring other 2-digit numbers

Let's say you want to square 58. Square each digit and write a partial answer. 5x5 = 25. 8x8 = 64.

Write down 2564 to start.

Then, multiply the two digits of the number you're squaring together, 5x8=40.

Double this product: 40x2=80, then add a 0 to it, getting 800.

Add 800 to 2564 to get 3364.

This is pretty complicated so let's do more examples.

32x32. The first part of the answer comes from squaring 3 and 2.

3x3=9. 2x2 = 4. Write down 0904. Notice the extra zeros. It's important that every square in the partial product have two digits.

Multiply the digits, 2 and 3, together and double the whole thing. 2x3x2 = 12.

Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.

56x56. The partial product comes from 5x5 and 6x6. Write down 2536.

5x6x2 = 60. Add a zero to get 600.

56x56 = 2536+600 = 3136.

One more example: 67x67. Write down 3649 as the partial product.

6x7x2 = 42x2 = 84. Add a zero to get 840.

67x67=3649+840 = 4489.

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Squaring 2-digit numbers that end in 5

If a number ends in 5 then its square always ends in 25.

To get the rest of the product take the left digit and multiply it by one more than itself.

35x35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3.

So, 3x4 = 12 and that's the rest of the product. Thus, 35x35 = 1225.

To calculate 65x65, notice that 6x7 = 42 and write down 4225 as the answer.

85x85: Calculate 8x9 = 72 and write down 7225.

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