x(x+5)=209

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Solution for x(x+5)=209 equation: 


Simplifying
x(x + 5) = 209

Reorder the terms:
x(5 + x) = 209
(5 * x + x * x) = 209
(5x + x2) = 209

Solving
5x + x2 = 209

Solving for variable 'x'.

Reorder the terms:
-209 + 5x + x2 = 209 + -209

Combine like terms: 209 + -209 = 0
-209 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '209' to each side of the equation.
-209 + 5x + 209 + x2 = 0 + 209

Reorder the terms:
-209 + 209 + 5x + x2 = 0 + 209

Combine like terms: -209 + 209 = 0
0 + 5x + x2 = 0 + 209
5x + x2 = 0 + 209

Combine like terms: 0 + 209 = 209
5x + x2 = 209

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 209 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 209 + 6.25

Combine like terms: 209 + 6.25 = 215.25
6.25 + 5x + x2 = 215.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 215.25

Calculate the square root of the right side: 14.671400751

Break this problem into two subproblems by setting 
(x + 2.5) equal to 14.671400751 and -14.671400751.


Subproblem 1
x + 2.5 = 14.671400751

Simplifying
x + 2.5 = 14.671400751

Reorder the terms:
2.5 + x = 14.671400751

Solving
2.5 + x = 14.671400751

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 14.671400751 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 14.671400751 + -2.5
x = 14.671400751 + -2.5

Combine like terms: 14.671400751 + -2.5 = 12.171400751
x = 12.171400751

Simplifying
x = 12.171400751


Subproblem 2
x + 2.5 = -14.671400751

Simplifying
x + 2.5 = -14.671400751

Reorder the terms:
2.5 + x = -14.671400751

Solving
2.5 + x = -14.671400751

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -14.671400751 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -14.671400751 + -2.5
x = -14.671400751 + -2.5

Combine like terms: -14.671400751 + -2.5 = -17.171400751
x = -17.171400751

Simplifying
x = -17.171400751


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.171400751, -17.171400751}

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