3x(5+3x)=9

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Solution for 3x(5+3x)=9 equation:


Simplifying
3x(5 + 3x) = 9
(5 * 3x + 3x * 3x) = 9
(15x + 9x2) = 9

Solving
15x + 9x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 15x + 9x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 15x + 9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + 5x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-3 + 5x + 3x2)' equal to zero and attempt to solve:

Simplifying
-3 + 5x + 3x2 = 0

Solving
-3 + 5x + 3x2 = 0

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

Divide each side by '3'.
-1 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.666666667x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.666666667x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.666666667x + x2 = 0 + 1
1.666666667x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.666666667x + x2 = 1

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 1 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 1 + 0.6944444447

Combine like terms: 1 + 0.6944444447 = 1.6944444447
0.6944444447 + 1.666666667x + x2 = 1.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 1.6944444447

Calculate the square root of the right side: 1.301708279

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 1.301708279 and -1.301708279.


Subproblem 1
x + 0.8333333335 = 1.301708279

Simplifying
x + 0.8333333335 = 1.301708279

Reorder the terms:
0.8333333335 + x = 1.301708279

Solving
0.8333333335 + x = 1.301708279

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 1.301708279 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 1.301708279 + -0.8333333335
x = 1.301708279 + -0.8333333335

Combine like terms: 1.301708279 + -0.8333333335 = 0.4683749455
x = 0.4683749455

Simplifying
x = 0.4683749455


Subproblem 2
x + 0.8333333335 = -1.301708279

Simplifying
x + 0.8333333335 = -1.301708279

Reorder the terms:
0.8333333335 + x = -1.301708279

Solving
0.8333333335 + x = -1.301708279

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -1.301708279 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -1.301708279 + -0.8333333335
x = -1.301708279 + -0.8333333335

Combine like terms: -1.301708279 + -0.8333333335 = -2.1350416125
x = -2.1350416125

Simplifying
x = -2.1350416125


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.4683749455, -2.1350416125}
Solution
x = {0.4683749455, -2.1350416125}

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