Given the squares 64x^2+a+121y^2 and 25x^4/16-b+16x^2/25 what are a and b.

If the sum 64x^2+a+121y^2 represents a perfect square, we'll
apply the formula:

(u + v)^2 = u^2 + 2uv +
v^2

We notice that the missing term is 2uv =
a.

We'll identify u^2 = 64x^2 => u = sqrt 64x^2 => u =
8x

v^2 = 121y^2 => v = sqrt 121y^2 => v =
11y

64x^2+a+121y^2

2uv =
2*8x*11y

2uv = 176xy

a =
176xy

The missing term in the quadratic expression is 176xy and the
completed square will be:

(8x+11y)^2 = 64x^2 + 176xy +
121y^2

We notice that the missing term is b = -2uv from the
formula:

(u - v)^2 = u^2 - 2uv +
v^2

We'll identify u^2 = 25x^4/16 => u = sqrt 25x^4/16
=> u = 5x^2/4

v^2 = 16x^2/25 => v = sqrt 16x^2/25
=> v =
-4x/5

25x^4/16-b+16x^2/25

-2uv =
-2*5x^2*4x/4*5

-2uv = -2x^3

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

(5x^2/4 - 4x/5)^2 = 25x^4/16- 2x^3 +
16x^2/25

The terms a and b are: a = 176xy and b =
2x^3.