# Fill in the squares 16x^4/36 - _ +36x^2/16= 81y^6 + 180y^3*x^5+_=

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1)16x^4/36 - ....... +36x^2/16=

To make the above a complete squre, we proceed as below:

We kow that (a-b)^2 = a^2+-2ab +b^2

We equate a^2 = (16x^4/36) = (4x^2/6)^2 . So a = 4x^2/6.

We equate b^2 = (36x^2/16. Therefore b = (6x/4).

Therefore the middle term -2ab = -2(4x^2/6)(6x/4) =

-288x^3/24.

Therefore (16x^4/36) - 288x^3/24 + (36x^2/16) = (4x^2/6 - 6x/4)^2.

2)

81y^6 + 180y^3*x^5+_= ? To complete this square we proceed as below:

We know that (a+b) ^2 = a^2 +2ab +b^2.

We equate a^2 = 81y^6 . So a= 9y^3.

We equate 2ab = 180y^3*x^5, where a = 9y^3 as found above.

So b = 180y^3x^5/2a = 180y^3x^5/(2*9y^3) = 10 x^5.

Therefore b= 10x^5.

Therefore (a+b)^2 = (9y^3+10x^5) = 81y^6+180y^3*x^5 +(10x^5)^5

(a+b)^2 = 81y^6+180y^3x^5 +100x^10

To complete the given squares, we'll have to work according to the formula:

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

We'll analyze the expression:

81y^6 + 180y^3*x^5+_=

We'll identify a^2 = 81y^6 => a = sqrt 81y^6

a = 9y^3

To calculate b, we'll consider the second term of the square:

180y^3*x^5 = 2*a*b

180y^3*x^5 = 2*9y^3*b

We'll use the symmetric property and we'll divide by -18y^3:

b = 180y^3*x^5/18y^3

b = 10x^5

Now, we'll complete the square by adding the amount b^2.

b^2 = ( 10x^5)^2

b^2 = 100x^10

(a+b)^2 = (9y^3 + 10x^5)^2

The missing term in the quadratic expression is 100x^10:

**(9y^3 + 10x^5)^2 = 81y^6 + 180y^3*x^5+100x^10**

2) We notice that the missing term is -2ab.

We'll identify a^2 = 16x^4/36 => a = sqrt 16x^4/36 => a = 4x^2/6

16x^4/36 - 2*4x^2/6*b + 36x^2/16 = 0

To calculate 2ab, we'll consider the 3rd term of the square:

36x^2/16 = b^2

b = sqrt 36x^2/16

b = -6x/4

2*4x^2/6*(-6x/4) = -2x^3

The missing term in the quadratic expression is -2x^3 and the completed square will be:

**(4x^2/6 - 6x/4)^2 = 16x^4/36-2x^3+36x^2/16**