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

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The expressions 64x^2 + a + 121y^2 and 25x^4/16- b + 16x^2/25 are squares.

(8x + 11y)^2 is a square and we can equate this to 64x^2 + a + 121y^2

=> (8x + 11y)^2 = 64x^2 + a + 121y^2

=> 64x^2 + 121y^2 + 2*8x*11y = 64x^2 + a + 121y^2

=> a = 176xy

Similarly.

[(5/4)x^2 - (4/5)x]^2 = 25x^4/16- b + 16x^2/25

=> (25/16)x^4 - 2*(4/5)(5/4)x^2*x + (16/25)x^2 = 25x^4/16- b + 16x^2/25

=> -b = -2x^3

=> b = 2x^3

**Therefore we get a = 176xy and b = 2x^3**

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.**