3x(2x+3)=45

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Solution for 3x(2x+3)=45 equation: 


Simplifying
3x(2x + 3) = 45

Reorder the terms:
3x(3 + 2x) = 45
(3 * 3x + 2x * 3x) = 45
(9x + 6x2) = 45

Solving
9x + 6x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 9x + 6x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 9x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-15 + 3x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-15 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-15 + 3x + 2x2 = 0

Solving
-15 + 3x + 2x2 = 0

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

Divide each side by '2'.
-7.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 1.5x + 7.5 + x2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 1.5x + x2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 1.5x + x2 = 0 + 7.5
1.5x + x2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
1.5x + x2 = 7.5

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 = 7.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 7.5 + 0.5625

Combine like terms: 7.5 + 0.5625 = 8.0625
0.5625 + 1.5x + x2 = 8.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(x + 0.75) equal to 2.839454173 and -2.839454173.


Subproblem 1
x + 0.75 = 2.839454173

Simplifying
x + 0.75 = 2.839454173

Reorder the terms:
0.75 + x = 2.839454173

Solving
0.75 + x = 2.839454173

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 = 2.839454173 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 2.839454173 + -0.75
x = 2.839454173 + -0.75

Combine like terms: 2.839454173 + -0.75 = 2.089454173
x = 2.089454173

Simplifying
x = 2.089454173


Subproblem 2
x + 0.75 = -2.839454173

Simplifying
x + 0.75 = -2.839454173

Reorder the terms:
0.75 + x = -2.839454173

Solving
0.75 + x = -2.839454173

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 = -2.839454173 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -2.839454173 + -0.75
x = -2.839454173 + -0.75

Combine like terms: -2.839454173 + -0.75 = -3.589454173
x = -3.589454173

Simplifying
x = -3.589454173


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.089454173, -3.589454173}
Solution
x = {2.089454173, -3.589454173}

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