-3x^2=14x-8

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Solution for -3x^2=14x-8 equation: 


Simplifying
-3x2 = 14x + -8

Reorder the terms:
-3x2 = -8 + 14x

Solving
-3x2 = -8 + 14x

Solving for variable 'x'.

Reorder the terms:
8 + -14x + -3x2 = -8 + 14x + 8 + -14x

Reorder the terms:
8 + -14x + -3x2 = -8 + 8 + 14x + -14x

Combine like terms: -8 + 8 = 0
8 + -14x + -3x2 = 0 + 14x + -14x
8 + -14x + -3x2 = 14x + -14x

Combine like terms: 14x + -14x = 0
8 + -14x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-2.666666667 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + 4.666666667x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + 4.666666667x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + 4.666666667x + x2 = 0 + 2.666666667
4.666666667x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
4.666666667x + x2 = 2.666666667

The x term is 4.666666667x.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667x + 5.444444448 + x2 = 2.666666667 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = 2.666666667 + 5.444444448

Combine like terms: 2.666666667 + 5.444444448 = 8.111111115
5.444444448 + 4.666666667x + x2 = 8.111111115

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 8.111111115

Calculate the square root of the right side: 2.848001249

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 2.848001249 and -2.848001249.


Subproblem 1
x + 2.333333334 = 2.848001249

Simplifying
x + 2.333333334 = 2.848001249

Reorder the terms:
2.333333334 + x = 2.848001249

Solving
2.333333334 + x = 2.848001249

Solving for variable 'x'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + x = 2.848001249 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + x = 2.848001249 + -2.333333334
x = 2.848001249 + -2.333333334

Combine like terms: 2.848001249 + -2.333333334 = 0.514667915
x = 0.514667915

Simplifying
x = 0.514667915


Subproblem 2
x + 2.333333334 = -2.848001249

Simplifying
x + 2.333333334 = -2.848001249

Reorder the terms:
2.333333334 + x = -2.848001249

Solving
2.333333334 + x = -2.848001249

Solving for variable 'x'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + x = -2.848001249 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + x = -2.848001249 + -2.333333334
x = -2.848001249 + -2.333333334

Combine like terms: -2.848001249 + -2.333333334 = -5.181334583
x = -5.181334583

Simplifying
x = -5.181334583


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.514667915, -5.181334583}

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