15x^2-x-54=0

Solution for 15x^2-x-54=0 equation:

Simplifying
15x2 + -1x + -54 = 0

Reorder the terms:
-54 + -1x + 15x2 = 0

Solving
-54 + -1x + 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'.
-3.6 + -0.06666666667x + x2 = 0

Move the constant term to the right:

Add '3.6' to each side of the equation.
-3.6 + -0.06666666667x + 3.6 + x2 = 0 + 3.6

Reorder the terms:
-3.6 + 3.6 + -0.06666666667x + x2 = 0 + 3.6

Combine like terms: -3.6 + 3.6 = 0.0
0.0 + -0.06666666667x + x2 = 0 + 3.6
-0.06666666667x + x2 = 0 + 3.6

Combine like terms: 0 + 3.6 = 3.6
-0.06666666667x + x2 = 3.6

The x term is -0.06666666667x.  Take half its coefficient (-0.03333333334).
Square it (0.001111111112) and add it to both sides.

Add '0.001111111112' to each side of the equation.
-0.06666666667x + 0.001111111112 + x2 = 3.6 + 0.001111111112

Reorder the terms:
0.001111111112 + -0.06666666667x + x2 = 3.6 + 0.001111111112

Combine like terms: 3.6 + 0.001111111112 = 3.601111111112
0.001111111112 + -0.06666666667x + x2 = 3.601111111112

Factor a perfect square on the left side:
(x + -0.03333333334)(x + -0.03333333334) = 3.601111111112

Calculate the square root of the right side: 1.897659377

Break this problem into two subproblems by setting 
(x + -0.03333333334) equal to 1.897659377 and -1.897659377.

Subproblem 1
x + -0.03333333334 = 1.897659377

Simplifying
x + -0.03333333334 = 1.897659377

Reorder the terms:
-0.03333333334 + x = 1.897659377

Solving
-0.03333333334 + x = 1.897659377

Solving for variable 'x'.

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

Add '0.03333333334' to each side of the equation.
-0.03333333334 + 0.03333333334 + x = 1.897659377 + 0.03333333334

Combine like terms: -0.03333333334 + 0.03333333334 = 0.00000000000
0.00000000000 + x = 1.897659377 + 0.03333333334
x = 1.897659377 + 0.03333333334

Combine like terms: 1.897659377 + 0.03333333334 = 1.93099271034
x = 1.93099271034

Simplifying
x = 1.93099271034

Subproblem 2
x + -0.03333333334 = -1.897659377

Simplifying
x + -0.03333333334 = -1.897659377

Reorder the terms:
-0.03333333334 + x = -1.897659377

Solving
-0.03333333334 + x = -1.897659377

Solving for variable 'x'.

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

Add '0.03333333334' to each side of the equation.
-0.03333333334 + 0.03333333334 + x = -1.897659377 + 0.03333333334

Combine like terms: -0.03333333334 + 0.03333333334 = 0.00000000000
0.00000000000 + x = -1.897659377 + 0.03333333334
x = -1.897659377 + 0.03333333334

Combine like terms: -1.897659377 + 0.03333333334 = -1.86432604366
x = -1.86432604366

Simplifying
x = -1.86432604366

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.93099271034, -1.86432604366}