0=x(x+1)-18

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Solution for 0=x(x+1)-18 equation:


Simplifying
0 = x(x + 1) + -18

Reorder the terms:
0 = x(1 + x) + -18
0 = (1 * x + x * x) + -18
0 = (1x + x2) + -18

Reorder the terms:
0 = -18 + 1x + x2

Solving
0 = -18 + 1x + x2

Solving for variable 'x'.

Combine like terms: 0 + 18 = 18
18 + -1x + -1x2 = -18 + 1x + x2 + 18 + -1x + -1x2

Reorder the terms:
18 + -1x + -1x2 = -18 + 18 + 1x + -1x + x2 + -1x2

Combine like terms: -18 + 18 = 0
18 + -1x + -1x2 = 0 + 1x + -1x + x2 + -1x2
18 + -1x + -1x2 = 1x + -1x + x2 + -1x2

Combine like terms: 1x + -1x = 0
18 + -1x + -1x2 = 0 + x2 + -1x2
18 + -1x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
18 + -1x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-18 + x + x2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + x + 18 + x2 = 0 + 18

Reorder the terms:
-18 + 18 + x + x2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + x + x2 = 0 + 18
x + x2 = 0 + 18

Combine like terms: 0 + 18 = 18
x + x2 = 18

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 18 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 18 + 0.25

Combine like terms: 18 + 0.25 = 18.25
1.25 + x2 = 18.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.272001873 and -4.272001873.

Subproblem 1

x + 0.5 = 4.272001873 Simplifying x + 0.5 = 4.272001873 Reorder the terms: 0.5 + x = 4.272001873 Solving 0.5 + x = 4.272001873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.272001873 + -0.5 x = 4.272001873 + -0.5 Combine like terms: 4.272001873 + -0.5 = 3.772001873 x = 3.772001873 Simplifying x = 3.772001873

Subproblem 2

x + 0.5 = -4.272001873 Simplifying x + 0.5 = -4.272001873 Reorder the terms: 0.5 + x = -4.272001873 Solving 0.5 + x = -4.272001873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -4.272001873 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.272001873 + -0.5 x = -4.272001873 + -0.5 Combine like terms: -4.272001873 + -0.5 = -4.772001873 x = -4.772001873 Simplifying x = -4.772001873

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.772001873, -4.772001873}

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