45x^2+120x+71=0

Solution for 45x^2+120x+71=0 equation:

Simplifying
45x2 + 120x + 71 = 0

Reorder the terms:
71 + 120x + 45x2 = 0

Solving
71 + 120x + 45x2 = 0

Solving for variable 'x'.

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

Divide each side by '45'.
1.577777778 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.577777778' to each side of the equation.
1.577777778 + 2.666666667x + -1.577777778 + x2 = 0 + -1.577777778

Reorder the terms:
1.577777778 + -1.577777778 + 2.666666667x + x2 = 0 + -1.577777778

Combine like terms: 1.577777778 + -1.577777778 = 0.000000000
0.000000000 + 2.666666667x + x2 = 0 + -1.577777778
2.666666667x + x2 = 0 + -1.577777778

Combine like terms: 0 + -1.577777778 = -1.577777778
2.666666667x + x2 = -1.577777778

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = -1.577777778 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = -1.577777778 + 1.777777780

Combine like terms: -1.577777778 + 1.777777780 = 0.200000002
1.777777780 + 2.666666667x + x2 = 0.200000002

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 0.200000002

Calculate the square root of the right side: 0.447213598

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 0.447213598 and -0.447213598.

Subproblem 1
x + 1.333333334 = 0.447213598

Simplifying
x + 1.333333334 = 0.447213598

Reorder the terms:
1.333333334 + x = 0.447213598

Solving
1.333333334 + x = 0.447213598

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = 0.447213598 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 0.447213598 + -1.333333334
x = 0.447213598 + -1.333333334

Combine like terms: 0.447213598 + -1.333333334 = -0.886119736
x = -0.886119736

Simplifying
x = -0.886119736

Subproblem 2
x + 1.333333334 = -0.447213598

Simplifying
x + 1.333333334 = -0.447213598

Reorder the terms:
1.333333334 + x = -0.447213598

Solving
1.333333334 + x = -0.447213598

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = -0.447213598 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -0.447213598 + -1.333333334
x = -0.447213598 + -1.333333334

Combine like terms: -0.447213598 + -1.333333334 = -1.780546932
x = -1.780546932

Simplifying
x = -1.780546932

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.886119736, -1.780546932}