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