6x+10+14x=5x(1+4x)

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Solution for 6x+10+14x=5x(1+4x) equation:


Simplifying
6x + 10 + 14x = 5x(1 + 4x)

Reorder the terms:
10 + 6x + 14x = 5x(1 + 4x)

Combine like terms: 6x + 14x = 20x
10 + 20x = 5x(1 + 4x)
10 + 20x = (1 * 5x + 4x * 5x)
10 + 20x = (5x + 20x2)

Solving
10 + 20x = 5x + 20x2

Solving for variable 'x'.

Combine like terms: 20x + -5x = 15x
10 + 15x + -20x2 = 5x + 20x2 + -5x + -20x2

Reorder the terms:
10 + 15x + -20x2 = 5x + -5x + 20x2 + -20x2

Combine like terms: 5x + -5x = 0
10 + 15x + -20x2 = 0 + 20x2 + -20x2
10 + 15x + -20x2 = 20x2 + -20x2

Combine like terms: 20x2 + -20x2 = 0
10 + 15x + -20x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(2 + 3x + -4x2) = 0

Ignore the factor 5.

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 = 1.17539053
x = 1.17539053

Simplifying
x = 1.17539053


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 = -0.42539053
x = -0.42539053

Simplifying
x = -0.42539053


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

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