4.5x^2+38x+31.875=0

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Solution for 4.5x^2+38x+31.875=0 equation:


Simplifying
4.5x2 + 38x + 31.875 = 0

Reorder the terms:
31.875 + 38x + 4.5x2 = 0

Solving
31.875 + 38x + 4.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '4.5'.
7.083333333 + 8.444444444x + x2 = 0

Move the constant term to the right:

Add '-7.083333333' to each side of the equation.
7.083333333 + 8.444444444x + -7.083333333 + x2 = 0 + -7.083333333

Reorder the terms:
7.083333333 + -7.083333333 + 8.444444444x + x2 = 0 + -7.083333333

Combine like terms: 7.083333333 + -7.083333333 = 0.000000000
0.000000000 + 8.444444444x + x2 = 0 + -7.083333333
8.444444444x + x2 = 0 + -7.083333333

Combine like terms: 0 + -7.083333333 = -7.083333333
8.444444444x + x2 = -7.083333333

The x term is 8.444444444x.  Take half its coefficient (4.222222222).
Square it (17.82716049) and add it to both sides.

Add '17.82716049' to each side of the equation.
8.444444444x + 17.82716049 + x2 = -7.083333333 + 17.82716049

Reorder the terms:
17.82716049 + 8.444444444x + x2 = -7.083333333 + 17.82716049

Combine like terms: -7.083333333 + 17.82716049 = 10.743827157
17.82716049 + 8.444444444x + x2 = 10.743827157

Factor a perfect square on the left side:
(x + 4.222222222)(x + 4.222222222) = 10.743827157

Calculate the square root of the right side: 3.277777777

Break this problem into two subproblems by setting 
(x + 4.222222222) equal to 3.277777777 and -3.277777777.


Subproblem 1
x + 4.222222222 = 3.277777777

Simplifying
x + 4.222222222 = 3.277777777

Reorder the terms:
4.222222222 + x = 3.277777777

Solving
4.222222222 + x = 3.277777777

Solving for variable 'x'.

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

Add '-4.222222222' to each side of the equation.
4.222222222 + -4.222222222 + x = 3.277777777 + -4.222222222

Combine like terms: 4.222222222 + -4.222222222 = 0.000000000
0.000000000 + x = 3.277777777 + -4.222222222
x = 3.277777777 + -4.222222222

Combine like terms: 3.277777777 + -4.222222222 = -0.944444445
x = -0.944444445

Simplifying
x = -0.944444445


Subproblem 2
x + 4.222222222 = -3.277777777

Simplifying
x + 4.222222222 = -3.277777777

Reorder the terms:
4.222222222 + x = -3.277777777

Solving
4.222222222 + x = -3.277777777

Solving for variable 'x'.

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

Add '-4.222222222' to each side of the equation.
4.222222222 + -4.222222222 + x = -3.277777777 + -4.222222222

Combine like terms: 4.222222222 + -4.222222222 = 0.000000000
0.000000000 + x = -3.277777777 + -4.222222222
x = -3.277777777 + -4.222222222

Combine like terms: -3.277777777 + -4.222222222 = -7.499999999
x = -7.499999999

Simplifying
x = -7.499999999


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.944444445, -7.499999999}

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