(2x+3)*4x-6=0

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Solution for (2x+3)*4x-6=0 equation: 


Simplifying
(2x + 3) * 4x + -6 = 0

Reorder the terms:
(3 + 2x) * 4x + -6 = 0

Reorder the terms for easier multiplication:
4x(3 + 2x) + -6 = 0
(3 * 4x + 2x * 4x) + -6 = 0
(12x + 8x2) + -6 = 0

Reorder the terms:
-6 + 12x + 8x2 = 0

Solving
-6 + 12x + 8x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 6x + 4x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-3 + 6x + 4x2)' equal to zero and attempt to solve:

Simplifying
-3 + 6x + 4x2 = 0

Solving
-3 + 6x + 4x2 = 0

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

Divide each side by '4'.
-0.75 + 1.5x + x2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.5x + 0.75 + x2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.5x + x2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.5x + x2 = 0 + 0.75
1.5x + x2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.5x + x2 = 0.75

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 = 0.75 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 0.75 + 0.5625

Combine like terms: 0.75 + 0.5625 = 1.3125
0.5625 + 1.5x + x2 = 1.3125

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(x + 0.75) equal to 1.145643924 and -1.145643924.


Subproblem 1
x + 0.75 = 1.145643924

Simplifying
x + 0.75 = 1.145643924

Reorder the terms:
0.75 + x = 1.145643924

Solving
0.75 + x = 1.145643924

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 = 1.145643924 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 1.145643924 + -0.75
x = 1.145643924 + -0.75

Combine like terms: 1.145643924 + -0.75 = 0.395643924
x = 0.395643924

Simplifying
x = 0.395643924


Subproblem 2
x + 0.75 = -1.145643924

Simplifying
x + 0.75 = -1.145643924

Reorder the terms:
0.75 + x = -1.145643924

Solving
0.75 + x = -1.145643924

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 = -1.145643924 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -1.145643924 + -0.75
x = -1.145643924 + -0.75

Combine like terms: -1.145643924 + -0.75 = -1.895643924
x = -1.895643924

Simplifying
x = -1.895643924


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.395643924, -1.895643924}
Solution
x = {0.395643924, -1.895643924}

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