6x(x+3)+1=31

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Solution for 6x(x+3)+1=31 equation: 


Simplifying
6x(x + 3) + 1 = 31

Reorder the terms:
6x(3 + x) + 1 = 31
(3 * 6x + x * 6x) + 1 = 31
(18x + 6x2) + 1 = 31

Reorder the terms:
1 + 18x + 6x2 = 31

Solving
1 + 18x + 6x2 = 31

Solving for variable 'x'.

Reorder the terms:
1 + -31 + 18x + 6x2 = 31 + -31

Combine like terms: 1 + -31 = -30
-30 + 18x + 6x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-30 + 18x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-5 + 3x + x2) = 0

Ignore the factor 6.

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

Simplifying
-5 + 3x + x2 = 0

Solving
-5 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 3x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 3x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 3x + x2 = 0 + 5
3x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
3x + x2 = 5

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3x + x2 = 7.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
x + 1.5 = 2.692582404

Simplifying
x + 1.5 = 2.692582404

Reorder the terms:
1.5 + x = 2.692582404

Solving
1.5 + x = 2.692582404

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.692582404 + -1.5
x = 2.692582404 + -1.5

Combine like terms: 2.692582404 + -1.5 = 1.192582404
x = 1.192582404

Simplifying
x = 1.192582404


Subproblem 2
x + 1.5 = -2.692582404

Simplifying
x + 1.5 = -2.692582404

Reorder the terms:
1.5 + x = -2.692582404

Solving
1.5 + x = -2.692582404

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.692582404 + -1.5
x = -2.692582404 + -1.5

Combine like terms: -2.692582404 + -1.5 = -4.192582404
x = -4.192582404

Simplifying
x = -4.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.192582404, -4.192582404}
Solution
x = {1.192582404, -4.192582404}

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