9j^2+3j-5=0

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Solution for 9j^2+3j-5=0 equation: 


Simplifying
9j2 + 3j + -5 = 0

Reorder the terms:
-5 + 3j + 9j2 = 0

Solving
-5 + 3j + 9j2 = 0

Solving for variable 'j'.

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

Divide each side by '9'.
-0.5555555556 + 0.3333333333j + j2 = 0

Move the constant term to the right:

Add '0.5555555556' to each side of the equation.
-0.5555555556 + 0.3333333333j + 0.5555555556 + j2 = 0 + 0.5555555556

Reorder the terms:
-0.5555555556 + 0.5555555556 + 0.3333333333j + j2 = 0 + 0.5555555556

Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000
0.0000000000 + 0.3333333333j + j2 = 0 + 0.5555555556
0.3333333333j + j2 = 0 + 0.5555555556

Combine like terms: 0 + 0.5555555556 = 0.5555555556
0.3333333333j + j2 = 0.5555555556

The j term is 0.3333333333j.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333j + 0.02777777779 + j2 = 0.5555555556 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333j + j2 = 0.5555555556 + 0.02777777779

Combine like terms: 0.5555555556 + 0.02777777779 = 0.58333333339
0.02777777779 + 0.3333333333j + j2 = 0.58333333339

Factor a perfect square on the left side:
(j + 0.1666666667)(j + 0.1666666667) = 0.58333333339

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(j + 0.1666666667) equal to 0.763762616 and -0.763762616.


Subproblem 1
j + 0.1666666667 = 0.763762616

Simplifying
j + 0.1666666667 = 0.763762616

Reorder the terms:
0.1666666667 + j = 0.763762616

Solving
0.1666666667 + j = 0.763762616

Solving for variable 'j'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + j = 0.763762616 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + j = 0.763762616 + -0.1666666667
j = 0.763762616 + -0.1666666667

Combine like terms: 0.763762616 + -0.1666666667 = 0.5970959493
j = 0.5970959493

Simplifying
j = 0.5970959493


Subproblem 2
j + 0.1666666667 = -0.763762616

Simplifying
j + 0.1666666667 = -0.763762616

Reorder the terms:
0.1666666667 + j = -0.763762616

Solving
0.1666666667 + j = -0.763762616

Solving for variable 'j'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + j = -0.763762616 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + j = -0.763762616 + -0.1666666667
j = -0.763762616 + -0.1666666667

Combine like terms: -0.763762616 + -0.1666666667 = -0.9304292827
j = -0.9304292827

Simplifying
j = -0.9304292827


Solution
The solution to the problem is based on the solutions
from the subproblems.
j = {0.5970959493, -0.9304292827}

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