9x^2+3x=5

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Solution for 9x^2+3x=5 equation:


Simplifying
9x2 + 3x = 5

Reorder the terms:
3x + 9x2 = 5

Solving
3x + 9x2 = 5

Solving for variable 'x'.

Reorder the terms:
-5 + 3x + 9x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 3x + 9x2 = 0

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

Divide each side by '9'.
-0.5555555556 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.5555555556' to each side of the equation.
-0.5555555556 + 0.3333333333x + 0.5555555556 + x2 = 0 + 0.5555555556

Reorder the terms:
-0.5555555556 + 0.5555555556 + 0.3333333333x + x2 = 0 + 0.5555555556

Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.5555555556
0.3333333333x + x2 = 0 + 0.5555555556

Combine like terms: 0 + 0.5555555556 = 0.5555555556
0.3333333333x + x2 = 0.5555555556

The x term is 0.3333333333x.  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.
0.3333333333x + 0.02777777779 + x2 = 0.5555555556 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.5555555556 + 0.02777777779

Combine like terms: 0.5555555556 + 0.02777777779 = 0.58333333339
0.02777777779 + 0.3333333333x + x2 = 0.58333333339

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

Calculate the square root of the right side: 0.763762616

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


Subproblem 1
x + 0.1666666667 = 0.763762616

Simplifying
x + 0.1666666667 = 0.763762616

Reorder the terms:
0.1666666667 + x = 0.763762616

Solving
0.1666666667 + x = 0.763762616

Solving for variable 'x'.

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

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

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

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

Simplifying
x = 0.5970959493


Subproblem 2
x + 0.1666666667 = -0.763762616

Simplifying
x + 0.1666666667 = -0.763762616

Reorder the terms:
0.1666666667 + x = -0.763762616

Solving
0.1666666667 + x = -0.763762616

Solving for variable 'x'.

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

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

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

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

Simplifying
x = -0.9304292827


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

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