21+(x+3)x=180

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Solution for 21+(x+3)x=180 equation: 


Simplifying
21 + (x + 3) * x = 180

Reorder the terms:
21 + (3 + x) * x = 180

Reorder the terms for easier multiplication:
21 + x(3 + x) = 180
21 + (3 * x + x * x) = 180
21 + (3x + x2) = 180

Solving
21 + 3x + x2 = 180

Solving for variable 'x'.

Reorder the terms:
21 + -180 + 3x + x2 = 180 + -180

Combine like terms: 21 + -180 = -159
-159 + 3x + x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-159 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '159' to each side of the equation.
-159 + 3x + 159 + x2 = 0 + 159

Reorder the terms:
-159 + 159 + 3x + x2 = 0 + 159

Combine like terms: -159 + 159 = 0
0 + 3x + x2 = 0 + 159
3x + x2 = 0 + 159

Combine like terms: 0 + 159 = 159
3x + x2 = 159

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 159 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 159 + 2.25

Combine like terms: 159 + 2.25 = 161.25
2.25 + 3x + x2 = 161.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 161.25

Calculate the square root of the right side: 12.698425099

Break this problem into two subproblems by setting 
(x + 1.5) equal to 12.698425099 and -12.698425099.


Subproblem 1
x + 1.5 = 12.698425099

Simplifying
x + 1.5 = 12.698425099

Reorder the terms:
1.5 + x = 12.698425099

Solving
1.5 + x = 12.698425099

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 12.698425099 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 12.698425099 + -1.5
x = 12.698425099 + -1.5

Combine like terms: 12.698425099 + -1.5 = 11.198425099
x = 11.198425099

Simplifying
x = 11.198425099


Subproblem 2
x + 1.5 = -12.698425099

Simplifying
x + 1.5 = -12.698425099

Reorder the terms:
1.5 + x = -12.698425099

Solving
1.5 + x = -12.698425099

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -12.698425099 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -12.698425099 + -1.5
x = -12.698425099 + -1.5

Combine like terms: -12.698425099 + -1.5 = -14.198425099
x = -14.198425099

Simplifying
x = -14.198425099


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.198425099, -14.198425099}

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