# Complete the squares x^2-14x+_ 9x^2-30x+_ 25x^2+_+36y^2 9x^4/25-_+25x^2/9 x^2+_+1/4

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1) To complete the given squares, we'll have to work according to the formula:

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

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

We'll analyze the first expression

x^2-14x+_

We'll identify a^2 = x^2 => a = x

To calculate b, we'll consider the second term of the square:

-14x = -2*a*b

-14x = -2*x*b

We'll use the symmetric property and we'll divide by 2x:

b = -7

Now, we'll complete the square by adding the amount b^2.

b^2 = (-7)^2

b^2 = 49

(a+b)^2 = (x-7)^2

The missing term in the quadratic expression is 49:

**(x-7)^2 = x^2 - 14x + 49**

2) We notice that the missing term is b^2.

We'll identify a^2 = 9x^2 => a = sqrt 9x^2 => a = 3x

9x^2-30x+_

To calculate b, we'll consider the second term of the square:

-30x = -2*a*b

-30x = -2*3x*b

-30x = -6x*b

We'll divide by 6x:

b = -5 => b^2 = 25

The missing term in the quadratic expression is 25 and the completed square will be:

**(3x-5)^2 = 9x^2-30x+25**

3) We notice that the missing term is 2ab.

We'll identify a^2 = 25x^2 => a = sqrt 25x^2 => a = 5x

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

25x^2+_+36y^2

2ab = 2*5x*6y

2ab = 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 2ab.

We'll identify a^2 = 9x^4/25 => a = sqrt 9x^4/25 => a = 3x^2/5

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

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

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

-2ab = -2x^3

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

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

5) We notice that the missing term is 2ab.

We'll identify a^2 = x^2 => a = sqrt x^2 => a = x

b^2 = 1/4 => b = sqrt 1/4 => b = 1/2

x^2+_+1/4

2ab = 2x/2

2ab = x

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

**(x + 1/2)^2 = x^2+ x +1/4**

Completing the the squares:

1) x^2-14x+_

x^2- 14x add (14/2)^2 = (x-7)^2

2)9x^2-30x+_

9x^2 -30x . Adding (30/2*3)^2 , we get (3x-5)^2 = 9x^2-30x+25

3)25x^2+_+36y^2:

25x^2+36x^2 by subteracting 2*5x*6y we get 25x^2-60xy+36y^2 = (5x-6y)^2.

4)9x^4/25+25x^2/9 by subtracting 2*(3x^2/5)(5x/3) we get 9x^4/25 - 2x^3 +25x^2/9 we get (3x^2/5 - 5x/3)^2

5)x^2+_+1/4 by subtracting -x we get (x-1/2)^2 .