3x^2-5=-13x

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Solution for 3x^2-5=-13x equation: 


Simplifying
3x2 + -5 = -13x

Reorder the terms:
-5 + 3x2 = -13x

Solving
-5 + 3x2 = -13x

Solving for variable 'x'.

Reorder the terms:
-5 + 13x + 3x2 = -13x + 13x

Combine like terms: -13x + 13x = 0
-5 + 13x + 3x2 = 0

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

Divide each side by '3'.
-1.666666667 + 4.333333333x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 4.333333333x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 4.333333333x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 4.333333333x + x2 = 0 + 1.666666667
4.333333333x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
4.333333333x + x2 = 1.666666667

The x term is 4.333333333x.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333x + 4.694444446 + x2 = 1.666666667 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333x + x2 = 1.666666667 + 4.694444446

Combine like terms: 1.666666667 + 4.694444446 = 6.361111113
4.694444446 + 4.333333333x + x2 = 6.361111113

Factor a perfect square on the left side:
(x + 2.166666667)(x + 2.166666667) = 6.361111113

Calculate the square root of the right side: 2.522124325

Break this problem into two subproblems by setting 
(x + 2.166666667) equal to 2.522124325 and -2.522124325.


Subproblem 1
x + 2.166666667 = 2.522124325

Simplifying
x + 2.166666667 = 2.522124325

Reorder the terms:
2.166666667 + x = 2.522124325

Solving
2.166666667 + x = 2.522124325

Solving for variable 'x'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = 2.522124325 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = 2.522124325 + -2.166666667
x = 2.522124325 + -2.166666667

Combine like terms: 2.522124325 + -2.166666667 = 0.355457658
x = 0.355457658

Simplifying
x = 0.355457658


Subproblem 2
x + 2.166666667 = -2.522124325

Simplifying
x + 2.166666667 = -2.522124325

Reorder the terms:
2.166666667 + x = -2.522124325

Solving
2.166666667 + x = -2.522124325

Solving for variable 'x'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + x = -2.522124325 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + x = -2.522124325 + -2.166666667
x = -2.522124325 + -2.166666667

Combine like terms: -2.522124325 + -2.166666667 = -4.688790992
x = -4.688790992

Simplifying
x = -4.688790992


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.355457658, -4.688790992}

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