Find a and b if 25x^2+a+36y^2 and 9x^4/25-b+25x^2/9 are squares.

If the sum 25x^2+a+36y^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 = 25x^2 => u = sqrt 25x^2
=> u = 5x

v^2 = 36y^2 => v = sqrt 36y^2
=> v = 6y

25x^2 + a +
36y^2

2uv = 2*5x*6y

2uv =
60xy

a = 60xy

The missing term
in the quadratic expression is 60xy and the completed square will
be:

(5x+6y)^2 = 25x^2 + 60xy +
36y^2

4) 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 = 9x^4/25 => u = sqrt 9x^4/25
=> u = 3x^2/5

v^2 = 25x^2/9 => v = sqrt
25x^2/9 => v = -5x/3

9x^4/25 - b +
25x^2/9

-2uv =
-2*3x^2*5x/5*3

-2uv =
-2x^3

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

(3x^2/5 - 5x/3)^2 = 9x^4/25- 2x^3 +
25x^2/9