15x^2+109x+8=0

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Solution for 15x^2+109x+8=0 equation:


Simplifying
15x2 + 109x + 8 = 0

Reorder the terms:
8 + 109x + 15x2 = 0

Solving
8 + 109x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
0.5333333333 + 7.266666667x + x2 = 0

Move the constant term to the right:

Add '-0.5333333333' to each side of the equation.
0.5333333333 + 7.266666667x + -0.5333333333 + x2 = 0 + -0.5333333333

Reorder the terms:
0.5333333333 + -0.5333333333 + 7.266666667x + x2 = 0 + -0.5333333333

Combine like terms: 0.5333333333 + -0.5333333333 = 0.0000000000
0.0000000000 + 7.266666667x + x2 = 0 + -0.5333333333
7.266666667x + x2 = 0 + -0.5333333333

Combine like terms: 0 + -0.5333333333 = -0.5333333333
7.266666667x + x2 = -0.5333333333

The x term is 7.266666667x.  Take half its coefficient (3.633333334).
Square it (13.20111112) and add it to both sides.

Add '13.20111112' to each side of the equation.
7.266666667x + 13.20111112 + x2 = -0.5333333333 + 13.20111112

Reorder the terms:
13.20111112 + 7.266666667x + x2 = -0.5333333333 + 13.20111112

Combine like terms: -0.5333333333 + 13.20111112 = 12.6677777867
13.20111112 + 7.266666667x + x2 = 12.6677777867

Factor a perfect square on the left side:
(x + 3.633333334)(x + 3.633333334) = 12.6677777867

Calculate the square root of the right side: 3.559182179

Break this problem into two subproblems by setting 
(x + 3.633333334) equal to 3.559182179 and -3.559182179.


Subproblem 1
x + 3.633333334 = 3.559182179

Simplifying
x + 3.633333334 = 3.559182179

Reorder the terms:
3.633333334 + x = 3.559182179

Solving
3.633333334 + x = 3.559182179

Solving for variable 'x'.

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

Add '-3.633333334' to each side of the equation.
3.633333334 + -3.633333334 + x = 3.559182179 + -3.633333334

Combine like terms: 3.633333334 + -3.633333334 = 0.000000000
0.000000000 + x = 3.559182179 + -3.633333334
x = 3.559182179 + -3.633333334

Combine like terms: 3.559182179 + -3.633333334 = -0.074151155
x = -0.074151155

Simplifying
x = -0.074151155


Subproblem 2
x + 3.633333334 = -3.559182179

Simplifying
x + 3.633333334 = -3.559182179

Reorder the terms:
3.633333334 + x = -3.559182179

Solving
3.633333334 + x = -3.559182179

Solving for variable 'x'.

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

Add '-3.633333334' to each side of the equation.
3.633333334 + -3.633333334 + x = -3.559182179 + -3.633333334

Combine like terms: 3.633333334 + -3.633333334 = 0.000000000
0.000000000 + x = -3.559182179 + -3.633333334
x = -3.559182179 + -3.633333334

Combine like terms: -3.559182179 + -3.633333334 = -7.192515513
x = -7.192515513

Simplifying
x = -7.192515513


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.074151155, -7.192515513}

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