(x^2+6x+9)+(3x+2)=0

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Solution for (x^2+6x+9)+(3x+2)=0 equation:


Simplifying
(x2 + 6x + 9) + (3x + 2) = 0

Reorder the terms:
(9 + 6x + x2) + (3x + 2) = 0

Remove parenthesis around (9 + 6x + x2)
9 + 6x + x2 + (3x + 2) = 0

Reorder the terms:
9 + 6x + x2 + (2 + 3x) = 0

Remove parenthesis around (2 + 3x)
9 + 6x + x2 + 2 + 3x = 0

Reorder the terms:
9 + 2 + 6x + 3x + x2 = 0

Combine like terms: 9 + 2 = 11
11 + 6x + 3x + x2 = 0

Combine like terms: 6x + 3x = 9x
11 + 9x + x2 = 0

Solving
11 + 9x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-11' to each side of the equation.
11 + 9x + -11 + x2 = 0 + -11

Reorder the terms:
11 + -11 + 9x + x2 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + 9x + x2 = 0 + -11
9x + x2 = 0 + -11

Combine like terms: 0 + -11 = -11
9x + x2 = -11

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = -11 + 20.25

Reorder the terms:
20.25 + 9x + x2 = -11 + 20.25

Combine like terms: -11 + 20.25 = 9.25
20.25 + 9x + x2 = 9.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x + 4.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

x + 4.5 = 3.041381265 Simplifying x + 4.5 = 3.041381265 Reorder the terms: 4.5 + x = 3.041381265 Solving 4.5 + x = 3.041381265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = 3.041381265 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = 3.041381265 + -4.5 x = 3.041381265 + -4.5 Combine like terms: 3.041381265 + -4.5 = -1.458618735 x = -1.458618735 Simplifying x = -1.458618735

Subproblem 2

x + 4.5 = -3.041381265 Simplifying x + 4.5 = -3.041381265 Reorder the terms: 4.5 + x = -3.041381265 Solving 4.5 + x = -3.041381265 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = -3.041381265 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = -3.041381265 + -4.5 x = -3.041381265 + -4.5 Combine like terms: -3.041381265 + -4.5 = -7.541381265 x = -7.541381265 Simplifying x = -7.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.458618735, -7.541381265}

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