x(1.5+x)=40

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Solution for x(1.5+x)=40 equation: 


Simplifying
x(1.5 + x) = 40
(1.5 * x + x * x) = 40
(1.5x + x2) = 40

Solving
1.5x + x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + 1.5x + x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 1.5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 1.5x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + 1.5x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 1.5x + x2 = 0 + 40
1.5x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
1.5x + x2 = 40

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 40 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 40 + 0.5625

Combine like terms: 40 + 0.5625 = 40.5625
0.5625 + 1.5x + x2 = 40.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 40.5625

Calculate the square root of the right side: 6.368869601

Break this problem into two subproblems by setting 
(x + 0.75) equal to 6.368869601 and -6.368869601.


Subproblem 1
x + 0.75 = 6.368869601

Simplifying
x + 0.75 = 6.368869601

Reorder the terms:
0.75 + x = 6.368869601

Solving
0.75 + x = 6.368869601

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 6.368869601 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 6.368869601 + -0.75
x = 6.368869601 + -0.75

Combine like terms: 6.368869601 + -0.75 = 5.618869601
x = 5.618869601

Simplifying
x = 5.618869601


Subproblem 2
x + 0.75 = -6.368869601

Simplifying
x + 0.75 = -6.368869601

Reorder the terms:
0.75 + x = -6.368869601

Solving
0.75 + x = -6.368869601

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -6.368869601 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -6.368869601 + -0.75
x = -6.368869601 + -0.75

Combine like terms: -6.368869601 + -0.75 = -7.118869601
x = -7.118869601

Simplifying
x = -7.118869601


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.618869601, -7.118869601}

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