Sum of Two Perfect Squares

Introduction to Sum of two perfect squares:

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.

(Source: Wikipedia)

Example for Sum of Two Perfect Squares:

Some of the examples for the sum of two perfect squares,

Example for sum of two perfect squares 1: `9, 100` find the sum of the perfect squares?

i) (9)

Solution:

`= 3^2`

`= 3 xx 3`

`= 9`

Therefore, 9 is a perfect square.

ii) 100 is a perfect square?

`= 10^2`

`= 10*10`

`= 100`

Therefore, `100 ` is a perfect square.

Therefore, sum of the perfect squares are,

`= 9 + 100`

`= 109`

Example for perfect square math 3: sqrt (64) is a perfect square?

`= 8 xx 8`

`= sqrt (64)`

Therefore, `sqrt (64)` is a perfect square.

Example for perfect square math 4: `(x + 6) ^2, (x + 5) ^2`

i) `(x +6)`

Solution: `= (x+6)2`

`= (x+6) (x+6)`

`= x(x+6) +6(x+6)`

`= x^2 + 6x + 6x + 64`

`= x^2 + 12x + 64`

There is another method also by using formula,

`= (x+6)^2 `

`= x^2 + 2 xx x xx 6 + 6^2 `

Therefore, `x^2+ 12x + 64 ` is a perfect square.

ii)` (x + 5)^2`

` = (x + 5) (x + 5)`

`= x(x + 5) + 5(x + 5)`

` = x^2 + 5x + 5x + 25`

` = x^2 + 10x + 25`

There is another method also by using formula,

`= (x + 5)^2`

`= x^2 + 2*x*5 + 5^2`

`= x^2 + 10x + 25`

Therefore, `x^2 + 10x + 25 ` is a perfect square.

Now, sum of those two perfect squares are taken,

`= (x^2+ 12x + 64)+ (x^2 + 10x + 25)`

` = 2x^2 + 22x + 99`

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Practice Problem for Perfect Square Math:

Practice problem for perfect square math 1: `38, 50`

Answer: `3944`

Practice problem for perfect square math 2: `(x+9) ^2, (x + 8) ^2`

Answer: `2x^2 + 34x + 145`