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

giorgiana1976 | Student

We'll recall the identities that helps us to complete the given squares:

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

(a-b)^2 = a^2 - 2ab + b^2

We'll complete the first square

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

To complete the 2nd aquare we'll identify what is the missing term and it 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

We'll complete the 3rd square. 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

We'll complete the 4th square. We'll identify the missing term as 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

We'll complete the fimal square. 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