3x^2+1.5x-46=181.5

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Solution for 3x^2+1.5x-46=181.5 equation: 


Simplifying
3x2 + 1.5x + -46 = 181.5

Reorder the terms:
-46 + 1.5x + 3x2 = 181.5

Solving
-46 + 1.5x + 3x2 = 181.5

Solving for variable 'x'.

Reorder the terms:
-46 + -181.5 + 1.5x + 3x2 = 181.5 + -181.5

Combine like terms: -46 + -181.5 = -227.5
-227.5 + 1.5x + 3x2 = 181.5 + -181.5

Combine like terms: 181.5 + -181.5 = 0.0
-227.5 + 1.5x + 3x2 = 0.0

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

Divide each side by '3'.
-75.83333333 + 0.5x + x2 = 0.0

Move the constant term to the right:

Add '75.83333333' to each side of the equation.
-75.83333333 + 0.5x + 75.83333333 + x2 = 0.0 + 75.83333333

Reorder the terms:
-75.83333333 + 75.83333333 + 0.5x + x2 = 0.0 + 75.83333333

Combine like terms: -75.83333333 + 75.83333333 = 0.00000000
0.00000000 + 0.5x + x2 = 0.0 + 75.83333333
0.5x + x2 = 0.0 + 75.83333333

Combine like terms: 0.0 + 75.83333333 = 75.83333333
0.5x + x2 = 75.83333333

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 75.83333333 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 75.83333333 + 0.0625

Combine like terms: 75.83333333 + 0.0625 = 75.89583333
0.0625 + 0.5x + x2 = 75.89583333

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 75.89583333

Calculate the square root of the right side: 8.71182147

Break this problem into two subproblems by setting 
(x + 0.25) equal to 8.71182147 and -8.71182147.


Subproblem 1
x + 0.25 = 8.71182147

Simplifying
x + 0.25 = 8.71182147

Reorder the terms:
0.25 + x = 8.71182147

Solving
0.25 + x = 8.71182147

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 8.71182147 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 8.71182147 + -0.25
x = 8.71182147 + -0.25

Combine like terms: 8.71182147 + -0.25 = 8.46182147
x = 8.46182147

Simplifying
x = 8.46182147


Subproblem 2
x + 0.25 = -8.71182147

Simplifying
x + 0.25 = -8.71182147

Reorder the terms:
0.25 + x = -8.71182147

Solving
0.25 + x = -8.71182147

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -8.71182147 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -8.71182147 + -0.25
x = -8.71182147 + -0.25

Combine like terms: -8.71182147 + -0.25 = -8.96182147
x = -8.96182147

Simplifying
x = -8.96182147


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.46182147, -8.96182147}

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