6x(5+2x)=54

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Solution for 6x(5+2x)=54 equation: 


Simplifying
6x(5 + 2x) = 54
(5 * 6x + 2x * 6x) = 54
(30x + 12x2) = 54

Solving
30x + 12x2 = 54

Solving for variable 'x'.

Reorder the terms:
-54 + 30x + 12x2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + 30x + 12x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-9 + 5x + 2x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-9 + 5x + 2x2)' equal to zero and attempt to solve:

Simplifying
-9 + 5x + 2x2 = 0

Solving
-9 + 5x + 2x2 = 0

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

Divide each side by '2'.
-4.5 + 2.5x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 2.5x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 2.5x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 2.5x + x2 = 0 + 4.5
2.5x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
2.5x + x2 = 4.5

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 4.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 4.5 + 1.5625

Combine like terms: 4.5 + 1.5625 = 6.0625
1.5625 + 2.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + 1.25) equal to 2.46221445 and -2.46221445.


Subproblem 1
x + 1.25 = 2.46221445

Simplifying
x + 1.25 = 2.46221445

Reorder the terms:
1.25 + x = 2.46221445

Solving
1.25 + x = 2.46221445

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = 2.46221445 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 2.46221445 + -1.25
x = 2.46221445 + -1.25

Combine like terms: 2.46221445 + -1.25 = 1.21221445
x = 1.21221445

Simplifying
x = 1.21221445


Subproblem 2
x + 1.25 = -2.46221445

Simplifying
x + 1.25 = -2.46221445

Reorder the terms:
1.25 + x = -2.46221445

Solving
1.25 + x = -2.46221445

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = -2.46221445 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -2.46221445 + -1.25
x = -2.46221445 + -1.25

Combine like terms: -2.46221445 + -1.25 = -3.71221445
x = -3.71221445

Simplifying
x = -3.71221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.21221445, -3.71221445}
Solution
x = {1.21221445, -3.71221445}

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