complete the square: (25/9)x^2 + (20/3)x + 20?

first of all the coefficient of x^2 must be 1
so multiply all the terms by (9/25)

x^2 + (12/5)x + (36/5)

X^2 + (12/5)x + (36/25) + (36/5) - (36/25)

(x + 6/5)^2 - (144/5)

364.4

(25 / 9)x² + (20 / 3)x + 20

Group.
(25 / 9)x² + (12 / 5)x] + 20

Factor
(25 / 9)[x² + (12 / 5)x] + 20

Add placeholders.
(25 / 9)[x² + (12 / 5)x + ___] + 20 - (25 / 9)(___)

Notice that the second blank is multiplied by -(25 / 9) to account for what you had to add to complete the square.

Take the coefficient of the x term: (12 / 5)
Divide it by 2: (12 / 5) / 2 = (12 / 10) = (6 / 5)
Square it: (6 / 5)² = (36 / 25)

Add (36 / 25) to both blanks.
(25 / 9)[x² + (12 / 5)x + (36 / 25)] + 20 - (25 / 9)(36 / 25)

x² + (12 / 5)x + (36 / 25) is the expanded form of a perfect square binomial.

Remember that (a + b)² = a² + 2ab + b². Apply this to what you have.
(25 / 9)[x² + (12 / 5)x + (36 / 25)] + 20 - (25 / 9)(36 / 25)
(25 / 9)[x + (6 / 5)]² + 20 - (25 / 9)(36 / 25)

Simplify the rest.
(25 / 9)[x + (6 / 5)]² + 20 - (25 / 9)(36 / 25)
(25 / 9)[x + (6 / 5)]² + 20 - (1 / 1)(4 / 1)
(25 / 9)[x + (6 / 5)]² + 20 - 4
(25 / 9)[x + (6 / 5)]² + 16

ANSWER: (25 / 9)[x + (6 / 5)]² + 16 is the vertex form.

BONUS: This means that the vertex is at (- 6 / 5, 16).
HINT: Remember that the vertex form is: y = a(x - h)² + k

CHECK:
(25 / 9)(x + (12 / 10)² + 16
(25 / 9)[(x)² + 2(x)(12 / 10)] + (12 / 10)²] + 16
(25 / 9)[x² + (12 / 5)x + (36 / 25)] + 16
(25 / 9)[x²] + (25 / 9)(12 / 5)x] + (25 / 9)(36 / 25)] + 16
(25 / 9)x² + (20 / 3)x + 4 + 16
(25 / 9)x² + (20 / 3)x + 20
TRUE

(25/9) x² + (20/3) x + 20 = 0
x² + (12/5) x = (- 9/25) 20
x² + (12/5) x = - 36/5
x² + (12/5) x + 36/25 = - 36/5 + 36/25
(x + 6/5)² = - 144/25
x + 6/5 = ± i √(144/25)
x + 6/5 = ± i (12/5)
x = - 6/5 ± i (12/5)
x = (6/5) (- 1 ± 2 i)

(25/9)x^2 + (20/3)x + 20
= x^2 + (20x/3)/(25/9) + 20/(25/9)
= x^2 + (20x/3)(9/25) + 20(9/25)
= x^2 + 12x/5 + 36/5
= x^2 + 6x/5 + 6x/5 + 36/5
= x^2 + 6x/5 + 6x/5 + 36/25 + 36/5 - 36/25
= (x^2 + 6x/5) + (6x/5 + 36/25) + 180/25 - 36/25
= x(x + 6/5) + 6/5(x + 6/5) + 144/25
= (x + 6/5)^2 + 144/25

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Answer
(1) ... ((5/3)x + 2)² + 16
............... OR
(2) ... (25/9)(x + 6/5)² + 16

We are given an expression in x. And, told to use the process of 'completing the square' to transform it into an expression having a perfect square.

Since the coefficient of x² is not 1, I can think of two approaches.
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____________________________________________________
1)
Given: (25/9)x² + (20/3)x + 20 ... complete the square.
Keeping 25/9 as the coefficient of x² , the answer will take the form
(1a) ... (ax+b)² + c ≡ a²x² + 2abx + b² + c
___________________________

'a' is obviously 5/3. So,
a = 5/3
_______
Now we need to determine the b that will get us (20/30)x as the middle term.
From (1a) we know the middle term will have the form 2abx so that
2abx = (20/3)x
2(5/3)b = (20/3)
b = 2
_______
Now
From (1a) we know the constant term will have the form (b²+c) and it will have to involve the constant term, 20, of the given expression.
So,
b² + c = 20
4 + c = 20
c = 16
_______
Substituting in for a, b, and c you get
((5/3)x+2)² + 16

That's the long of it.
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Here's the summation

(25/9)x² + (20/3)x + 20
= (5/3)²x² + 2(5/3)(2)x + 4 + 16
= [(5/3)²x² + 2(5/3)(2)x + 4] + 16
= ((5/3)x + 2)² + 16

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_______________________________________________
2)
Given: (25/9)x² + (20/3)x + 20 ... complete the square.
Factor out 25/9 to get the coefficient of x² to 1.

(25/9)x² + (20/3)x + 20
= (25/9)[x² + (12/5)x] + 20
= (25/9)[x² + (12/5)x + (6/5)² - (6/5)²] + 20
= (25/9)[x² + (12/5)x + 36/25] - (25/9)(6/5)² + 20
= (25/9)(x + 6/5)² - 4 + 20
= (25/9)(x + 6/5)² + 16

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I get:

(25/9)x^2 + 20/3 + 20 = 0.....25/9 is a square, and 20/3 is compatible, so....

25/9)x^2 + 20/3 + 1 + 19 -19 = -19

[(5/3)x + 1]^2 = -19

(5/3)x + 1 = ±√-19

(5/3)x = -1±√-19

5x = -3±√-19

x = -3/5 ±√-19......answer