14-3x(4-x)=28+5x(3x-8)

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Solution for 14-3x(4-x)=28+5x(3x-8) equation: 


Simplifying
14 + -3x(4 + -1x) = 28 + 5x(3x + -8)
14 + (4 * -3x + -1x * -3x) = 28 + 5x(3x + -8)
14 + (-12x + 3x2) = 28 + 5x(3x + -8)

Reorder the terms:
14 + -12x + 3x2 = 28 + 5x(-8 + 3x)
14 + -12x + 3x2 = 28 + (-8 * 5x + 3x * 5x)
14 + -12x + 3x2 = 28 + (-40x + 15x2)

Solving
14 + -12x + 3x2 = 28 + -40x + 15x2

Solving for variable 'x'.

Reorder the terms:
14 + -28 + -12x + 40x + 3x2 + -15x2 = 28 + -40x + 15x2 + -28 + 40x + -15x2

Combine like terms: 14 + -28 = -14
-14 + -12x + 40x + 3x2 + -15x2 = 28 + -40x + 15x2 + -28 + 40x + -15x2

Combine like terms: -12x + 40x = 28x
-14 + 28x + 3x2 + -15x2 = 28 + -40x + 15x2 + -28 + 40x + -15x2

Combine like terms: 3x2 + -15x2 = -12x2
-14 + 28x + -12x2 = 28 + -40x + 15x2 + -28 + 40x + -15x2

Reorder the terms:
-14 + 28x + -12x2 = 28 + -28 + -40x + 40x + 15x2 + -15x2

Combine like terms: 28 + -28 = 0
-14 + 28x + -12x2 = 0 + -40x + 40x + 15x2 + -15x2
-14 + 28x + -12x2 = -40x + 40x + 15x2 + -15x2

Combine like terms: -40x + 40x = 0
-14 + 28x + -12x2 = 0 + 15x2 + -15x2
-14 + 28x + -12x2 = 15x2 + -15x2

Combine like terms: 15x2 + -15x2 = 0
-14 + 28x + -12x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + 14x + -6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-7 + 14x + -6x2)' equal to zero and attempt to solve:

Simplifying
-7 + 14x + -6x2 = 0

Solving
-7 + 14x + -6x2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
1.166666667 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '-1.166666667' to each side of the equation.
1.166666667 + -2.333333333x + -1.166666667 + x2 = 0 + -1.166666667

Reorder the terms:
1.166666667 + -1.166666667 + -2.333333333x + x2 = 0 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + -2.333333333x + x2 = 0 + -1.166666667
-2.333333333x + x2 = 0 + -1.166666667

Combine like terms: 0 + -1.166666667 = -1.166666667
-2.333333333x + x2 = -1.166666667

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = -1.166666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = -1.166666667 + 1.361111112

Combine like terms: -1.166666667 + 1.361111112 = 0.194444445
1.361111112 + -2.333333333x + x2 = 0.194444445

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 0.194444445

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 0.440958552 and -0.440958552.


Subproblem 1
x + -1.166666667 = 0.440958552

Simplifying
x + -1.166666667 = 0.440958552

Reorder the terms:
-1.166666667 + x = 0.440958552

Solving
-1.166666667 + x = 0.440958552

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 0.440958552 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 0.440958552 + 1.166666667
x = 0.440958552 + 1.166666667

Combine like terms: 0.440958552 + 1.166666667 = 1.607625219
x = 1.607625219

Simplifying
x = 1.607625219


Subproblem 2
x + -1.166666667 = -0.440958552

Simplifying
x + -1.166666667 = -0.440958552

Reorder the terms:
-1.166666667 + x = -0.440958552

Solving
-1.166666667 + x = -0.440958552

Solving for variable 'x'.

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

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -0.440958552 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -0.440958552 + 1.166666667
x = -0.440958552 + 1.166666667

Combine like terms: -0.440958552 + 1.166666667 = 0.725708115
x = 0.725708115

Simplifying
x = 0.725708115


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.607625219, 0.725708115}
Solution
x = {1.607625219, 0.725708115}

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