5x^2+9x=-x^2+5x+1

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Solution for 5x^2+9x=-x^2+5x+1 equation: 


Simplifying
5x2 + 9x = -1x2 + 5x + 1

Reorder the terms:
9x + 5x2 = -1x2 + 5x + 1

Reorder the terms:
9x + 5x2 = 1 + 5x + -1x2

Solving
9x + 5x2 = 1 + 5x + -1x2

Solving for variable 'x'.

Reorder the terms:
-1 + 9x + -5x + 5x2 + x2 = 1 + 5x + -1x2 + -1 + -5x + x2

Combine like terms: 9x + -5x = 4x
-1 + 4x + 5x2 + x2 = 1 + 5x + -1x2 + -1 + -5x + x2

Combine like terms: 5x2 + x2 = 6x2
-1 + 4x + 6x2 = 1 + 5x + -1x2 + -1 + -5x + x2

Reorder the terms:
-1 + 4x + 6x2 = 1 + -1 + 5x + -5x + -1x2 + x2

Combine like terms: 1 + -1 = 0
-1 + 4x + 6x2 = 0 + 5x + -5x + -1x2 + x2
-1 + 4x + 6x2 = 5x + -5x + -1x2 + x2

Combine like terms: 5x + -5x = 0
-1 + 4x + 6x2 = 0 + -1x2 + x2
-1 + 4x + 6x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-1 + 4x + 6x2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-0.1666666667 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.6666666667x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 0.6666666667x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 0.6666666667x + x2 = 0 + 0.1666666667
0.6666666667x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
0.6666666667x + x2 = 0.1666666667

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 0.1666666667 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 0.1666666667 + 0.1111111112

Combine like terms: 0.1666666667 + 0.1111111112 = 0.2777777779
0.1111111112 + 0.6666666667x + x2 = 0.2777777779

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 0.2777777779

Calculate the square root of the right side: 0.527046277

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 0.527046277 and -0.527046277.


Subproblem 1
x + 0.3333333334 = 0.527046277

Simplifying
x + 0.3333333334 = 0.527046277

Reorder the terms:
0.3333333334 + x = 0.527046277

Solving
0.3333333334 + x = 0.527046277

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 0.527046277 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 0.527046277 + -0.3333333334
x = 0.527046277 + -0.3333333334

Combine like terms: 0.527046277 + -0.3333333334 = 0.1937129436
x = 0.1937129436

Simplifying
x = 0.1937129436


Subproblem 2
x + 0.3333333334 = -0.527046277

Simplifying
x + 0.3333333334 = -0.527046277

Reorder the terms:
0.3333333334 + x = -0.527046277

Solving
0.3333333334 + x = -0.527046277

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -0.527046277 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -0.527046277 + -0.3333333334
x = -0.527046277 + -0.3333333334

Combine like terms: -0.527046277 + -0.3333333334 = -0.8603796104
x = -0.8603796104

Simplifying
x = -0.8603796104


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.1937129436, -0.8603796104}

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