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

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