3x(2x+1)+1=34

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Solution for 3x(2x+1)+1=34 equation: 


Simplifying
3x(2x + 1) + 1 = 34

Reorder the terms:
3x(1 + 2x) + 1 = 34
(1 * 3x + 2x * 3x) + 1 = 34
(3x + 6x2) + 1 = 34

Reorder the terms:
1 + 3x + 6x2 = 34

Solving
1 + 3x + 6x2 = 34

Solving for variable 'x'.

Reorder the terms:
1 + -34 + 3x + 6x2 = 34 + -34

Combine like terms: 1 + -34 = -33
-33 + 3x + 6x2 = 34 + -34

Combine like terms: 34 + -34 = 0
-33 + 3x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-11 + x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-11 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-11 + x + 2x2 = 0

Solving
-11 + x + 2x2 = 0

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

Divide each side by '2'.
-5.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 0.5x + 5.5 + x2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 0.5x + x2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 0.5x + x2 = 0 + 5.5
0.5x + x2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
0.5x + x2 = 5.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 5.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 5.5 + 0.25

Combine like terms: 5.5 + 0.25 = 5.75
0.25 + 0.5x + x2 = 5.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.397915762 and -2.397915762.


Subproblem 1
x + 0.5 = 2.397915762

Simplifying
x + 0.5 = 2.397915762

Reorder the terms:
0.5 + x = 2.397915762

Solving
0.5 + x = 2.397915762

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.397915762 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.397915762 + -0.5
x = 2.397915762 + -0.5

Combine like terms: 2.397915762 + -0.5 = 1.897915762
x = 1.897915762

Simplifying
x = 1.897915762


Subproblem 2
x + 0.5 = -2.397915762

Simplifying
x + 0.5 = -2.397915762

Reorder the terms:
0.5 + x = -2.397915762

Solving
0.5 + x = -2.397915762

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.397915762 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.397915762 + -0.5
x = -2.397915762 + -0.5

Combine like terms: -2.397915762 + -0.5 = -2.897915762
x = -2.897915762

Simplifying
x = -2.897915762


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.897915762, -2.897915762}
Solution
x = {1.897915762, -2.897915762}

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