x+x(x+2)=51

Solution for x+x(x+2)=51 equation:

Simplifying
x + x(x + 2) = 51

Reorder the terms:
x + x(2 + x) = 51
x + (2 * x + x * x) = 51
x + (2x + x2) = 51

Combine like terms: x + 2x = 3x
3x + x2 = 51

Solving
3x + x2 = 51

Solving for variable 'x'.

Reorder the terms:
-51 + 3x + x2 = 51 + -51

Combine like terms: 51 + -51 = 0
-51 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '51' to each side of the equation.
-51 + 3x + 51 + x2 = 0 + 51

Reorder the terms:
-51 + 51 + 3x + x2 = 0 + 51

Combine like terms: -51 + 51 = 0
0 + 3x + x2 = 0 + 51
3x + x2 = 0 + 51

Combine like terms: 0 + 51 = 51
3x + x2 = 51

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 = 51 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 51 + 2.25

Combine like terms: 51 + 2.25 = 53.25
2.25 + 3x + x2 = 53.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 53.25

Calculate the square root of the right side: 7.29725976

Break this problem into two subproblems by setting 
(x + 1.5) equal to 7.29725976 and -7.29725976.

Subproblem 1
x + 1.5 = 7.29725976

Simplifying
x + 1.5 = 7.29725976

Reorder the terms:
1.5 + x = 7.29725976

Solving
1.5 + x = 7.29725976

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 = 7.29725976 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 7.29725976 + -1.5
x = 7.29725976 + -1.5

Combine like terms: 7.29725976 + -1.5 = 5.79725976
x = 5.79725976

Simplifying
x = 5.79725976

Subproblem 2
x + 1.5 = -7.29725976

Simplifying
x + 1.5 = -7.29725976

Reorder the terms:
1.5 + x = -7.29725976

Solving
1.5 + x = -7.29725976

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 = -7.29725976 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -7.29725976 + -1.5
x = -7.29725976 + -1.5

Combine like terms: -7.29725976 + -1.5 = -8.79725976
x = -8.79725976

Simplifying
x = -8.79725976

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.79725976, -8.79725976}