5x(9+6x)=9

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Solution for 5x(9+6x)=9 equation: 


Simplifying
5x(9 + 6x) = 9
(9 * 5x + 6x * 5x) = 9
(45x + 30x2) = 9

Solving
45x + 30x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 45x + 30x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 45x + 30x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + 15x + 10x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-3 + 15x + 10x2)' equal to zero and attempt to solve:

Simplifying
-3 + 15x + 10x2 = 0

Solving
-3 + 15x + 10x2 = 0

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

Divide each side by '10'.
-0.3 + 1.5x + x2 = 0.0

Move the constant term to the right:

Add '0.3' to each side of the equation.
-0.3 + 1.5x + 0.3 + x2 = 0.0 + 0.3

Reorder the terms:
-0.3 + 0.3 + 1.5x + x2 = 0.0 + 0.3

Combine like terms: -0.3 + 0.3 = 0.0
0.0 + 1.5x + x2 = 0.0 + 0.3
1.5x + x2 = 0.0 + 0.3

Combine like terms: 0.0 + 0.3 = 0.3
1.5x + x2 = 0.3

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 0.3 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 0.3 + 0.5625

Combine like terms: 0.3 + 0.5625 = 0.8625
0.5625 + 1.5x + x2 = 0.8625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 0.8625

Calculate the square root of the right side: 0.928708781

Break this problem into two subproblems by setting 
(x + 0.75) equal to 0.928708781 and -0.928708781.


Subproblem 1
x + 0.75 = 0.928708781

Simplifying
x + 0.75 = 0.928708781

Reorder the terms:
0.75 + x = 0.928708781

Solving
0.75 + x = 0.928708781

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 0.928708781 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 0.928708781 + -0.75
x = 0.928708781 + -0.75

Combine like terms: 0.928708781 + -0.75 = 0.178708781
x = 0.178708781

Simplifying
x = 0.178708781


Subproblem 2
x + 0.75 = -0.928708781

Simplifying
x + 0.75 = -0.928708781

Reorder the terms:
0.75 + x = -0.928708781

Solving
0.75 + x = -0.928708781

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -0.928708781 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -0.928708781 + -0.75
x = -0.928708781 + -0.75

Combine like terms: -0.928708781 + -0.75 = -1.678708781
x = -1.678708781

Simplifying
x = -1.678708781


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.178708781, -1.678708781}
Solution
x = {0.178708781, -1.678708781}

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