# Find 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 have to determine a and b.

25x^2+a+36y^2

=> (5x)^2 + a + (6y)^2

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

So we can say a = 2*5x*6y = 60xy

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

=> (3x^2/5)^2 - b + (5x/3)^2

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

So we can say that b = -2*(3x^2/5)*(5x/3)

=> b = -2*3*x^2*5/(5x*3)

=> b = -2*x^3

**Therefore a = 60xy and b = -2x^3**.

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**