2x+3=x^2-10x-5

Solution for 2x+3=x^2-10x-5 equation:

Simplifying
2x + 3 = x2 + -10x + -5

Reorder the terms:
3 + 2x = x2 + -10x + -5

Reorder the terms:
3 + 2x = -5 + -10x + x2

Solving
3 + 2x = -5 + -10x + x2

Solving for variable 'x'.

Reorder the terms:
3 + 5 + 2x + 10x + -1x2 = -5 + -10x + x2 + 5 + 10x + -1x2

Combine like terms: 3 + 5 = 8
8 + 2x + 10x + -1x2 = -5 + -10x + x2 + 5 + 10x + -1x2

Combine like terms: 2x + 10x = 12x
8 + 12x + -1x2 = -5 + -10x + x2 + 5 + 10x + -1x2

Reorder the terms:
8 + 12x + -1x2 = -5 + 5 + -10x + 10x + x2 + -1x2

Combine like terms: -5 + 5 = 0
8 + 12x + -1x2 = 0 + -10x + 10x + x2 + -1x2
8 + 12x + -1x2 = -10x + 10x + x2 + -1x2

Combine like terms: -10x + 10x = 0
8 + 12x + -1x2 = 0 + x2 + -1x2
8 + 12x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
8 + 12x + -1x2 = 0

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

Divide each side by '-1'.
-8 + -12x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -12x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + -12x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -12x + x2 = 0 + 8
-12x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
-12x + x2 = 8

The x term is -12x.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12x + 36 + x2 = 8 + 36

Reorder the terms:
36 + -12x + x2 = 8 + 36

Combine like terms: 8 + 36 = 44
36 + -12x + x2 = 44

Factor a perfect square on the left side:
(x + -6)(x + -6) = 44

Calculate the square root of the right side: 6.633249581

Break this problem into two subproblems by setting 
(x + -6) equal to 6.633249581 and -6.633249581.

Subproblem 1
x + -6 = 6.633249581

Simplifying
x + -6 = 6.633249581

Reorder the terms:
-6 + x = 6.633249581

Solving
-6 + x = 6.633249581

Solving for variable 'x'.

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

Add '6' to each side of the equation.
-6 + 6 + x = 6.633249581 + 6

Combine like terms: -6 + 6 = 0
0 + x = 6.633249581 + 6
x = 6.633249581 + 6

Combine like terms: 6.633249581 + 6 = 12.633249581
x = 12.633249581

Simplifying
x = 12.633249581

Subproblem 2
x + -6 = -6.633249581

Simplifying
x + -6 = -6.633249581

Reorder the terms:
-6 + x = -6.633249581

Solving
-6 + x = -6.633249581

Solving for variable 'x'.

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

Add '6' to each side of the equation.
-6 + 6 + x = -6.633249581 + 6

Combine like terms: -6 + 6 = 0
0 + x = -6.633249581 + 6
x = -6.633249581 + 6

Combine like terms: -6.633249581 + 6 = -0.633249581
x = -0.633249581

Simplifying
x = -0.633249581

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.633249581, -0.633249581}