6x(3+4x)=12

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Solution for 6x(3+4x)=12 equation:


Simplifying
6x(3 + 4x) = 12
(3 * 6x + 4x * 6x) = 12
(18x + 24x2) = 12

Solving
18x + 24x2 = 12

Solving for variable 'x'.

Reorder the terms:
-12 + 18x + 24x2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 18x + 24x2 = 0

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

Ignore the factor 6.

Subproblem 1
Set the factor '(-2 + 3x + 4x2)' equal to zero and attempt to solve:

Simplifying
-2 + 3x + 4x2 = 0

Solving
-2 + 3x + 4x2 = 0

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

Divide each side by '4'.
-0.5 + 0.75x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.75x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.75x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.75x + x2 = 0 + 0.5
0.75x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.75x + x2 = 0.5

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + 0.75x + x2 = 0.640625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(x + 0.375) equal to 0.80039053 and -0.80039053.


Subproblem 1
x + 0.375 = 0.80039053

Simplifying
x + 0.375 = 0.80039053

Reorder the terms:
0.375 + x = 0.80039053

Solving
0.375 + x = 0.80039053

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 0.80039053 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 0.80039053 + -0.375
x = 0.80039053 + -0.375

Combine like terms: 0.80039053 + -0.375 = 0.42539053
x = 0.42539053

Simplifying
x = 0.42539053


Subproblem 2
x + 0.375 = -0.80039053

Simplifying
x + 0.375 = -0.80039053

Reorder the terms:
0.375 + x = -0.80039053

Solving
0.375 + x = -0.80039053

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -0.80039053 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -0.80039053 + -0.375
x = -0.80039053 + -0.375

Combine like terms: -0.80039053 + -0.375 = -1.17539053
x = -1.17539053

Simplifying
x = -1.17539053


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.42539053, -1.17539053}
Solution
x = {0.42539053, -1.17539053}

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