14x+3x^2-11=0

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Solution for 14x+3x^2-11=0 equation: 


Simplifying
14x + 3x2 + -11 = 0

Reorder the terms:
-11 + 14x + 3x2 = 0

Solving
-11 + 14x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-3.666666667 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + 4.666666667x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + 4.666666667x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + 4.666666667x + x2 = 0 + 3.666666667
4.666666667x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
4.666666667x + x2 = 3.666666667

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 = 3.666666667 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = 3.666666667 + 5.444444448

Combine like terms: 3.666666667 + 5.444444448 = 9.111111115
5.444444448 + 4.666666667x + x2 = 9.111111115

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 9.111111115

Calculate the square root of the right side: 3.018461713

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 3.018461713 and -3.018461713.


Subproblem 1
x + 2.333333334 = 3.018461713

Simplifying
x + 2.333333334 = 3.018461713

Reorder the terms:
2.333333334 + x = 3.018461713

Solving
2.333333334 + x = 3.018461713

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 = 3.018461713 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + x = 3.018461713 + -2.333333334
x = 3.018461713 + -2.333333334

Combine like terms: 3.018461713 + -2.333333334 = 0.685128379
x = 0.685128379

Simplifying
x = 0.685128379


Subproblem 2
x + 2.333333334 = -3.018461713

Simplifying
x + 2.333333334 = -3.018461713

Reorder the terms:
2.333333334 + x = -3.018461713

Solving
2.333333334 + x = -3.018461713

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 = -3.018461713 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + x = -3.018461713 + -2.333333334
x = -3.018461713 + -2.333333334

Combine like terms: -3.018461713 + -2.333333334 = -5.351795047
x = -5.351795047

Simplifying
x = -5.351795047


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.685128379, -5.351795047}

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