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

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We are given that 25x^2+a+36y^2 and 9x^4/25-b+25x^2/9 are squares and we need to determine a and b.

The two can be squares of the form (a - b)^2 and (a + b)^2

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

a can be 2*5x*6y or -2*5x*6y

a = 60xy or x = -60xy

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

b can be 2*(3x^2/5)(5x/3) or -2(3x^2/5)(5x/3)

b = 2x^3 or -2x^3

**The required values of a can be {60xy, -60xy} and the values of b can be {2x^3, -2x^3}**

Since 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 the squares: 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**

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

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

We'll identify the squares: 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**