5x+6x^2-19=0

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Solution for 5x+6x^2-19=0 equation: 


Simplifying
5x + 6x2 + -19 = 0

Reorder the terms:
-19 + 5x + 6x2 = 0

Solving
-19 + 5x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.166666667 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + 0.8333333333x + 3.166666667 + x2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + 0.8333333333x + x2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + 0.8333333333x + x2 = 0 + 3.166666667
0.8333333333x + x2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
0.8333333333x + x2 = 3.166666667

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = 3.166666667 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 3.166666667 + 0.1736111111

Combine like terms: 3.166666667 + 0.1736111111 = 3.3402777781
0.1736111111 + 0.8333333333x + x2 = 3.3402777781

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 3.3402777781

Calculate the square root of the right side: 1.827642683

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 1.827642683 and -1.827642683.


Subproblem 1
x + 0.4166666667 = 1.827642683

Simplifying
x + 0.4166666667 = 1.827642683

Reorder the terms:
0.4166666667 + x = 1.827642683

Solving
0.4166666667 + x = 1.827642683

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = 1.827642683 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = 1.827642683 + -0.4166666667
x = 1.827642683 + -0.4166666667

Combine like terms: 1.827642683 + -0.4166666667 = 1.4109760163
x = 1.4109760163

Simplifying
x = 1.4109760163


Subproblem 2
x + 0.4166666667 = -1.827642683

Simplifying
x + 0.4166666667 = -1.827642683

Reorder the terms:
0.4166666667 + x = -1.827642683

Solving
0.4166666667 + x = -1.827642683

Solving for variable 'x'.

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

Add '-0.4166666667' to each side of the equation.
0.4166666667 + -0.4166666667 + x = -1.827642683 + -0.4166666667

Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000
0.0000000000 + x = -1.827642683 + -0.4166666667
x = -1.827642683 + -0.4166666667

Combine like terms: -1.827642683 + -0.4166666667 = -2.2443093497
x = -2.2443093497

Simplifying
x = -2.2443093497


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4109760163, -2.2443093497}

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