x^2+x=106

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Solution for x^2+x=106 equation:


Simplifying
x2 + x = 106

Reorder the terms:
x + x2 = 106

Solving
x + x2 = 106

Solving for variable 'x'.

Reorder the terms:
-106 + x + x2 = 106 + -106

Combine like terms: 106 + -106 = 0
-106 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '106' to each side of the equation.
-106 + x + 106 + x2 = 0 + 106

Reorder the terms:
-106 + 106 + x + x2 = 0 + 106

Combine like terms: -106 + 106 = 0
0 + x + x2 = 0 + 106
x + x2 = 0 + 106

Combine like terms: 0 + 106 = 106
x + x2 = 106

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.
x + 0.25 + x2 = 106 + 0.25

Reorder the terms:
0.25 + x + x2 = 106 + 0.25

Combine like terms: 106 + 0.25 = 106.25
0.25 + x + x2 = 106.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 106.25

Calculate the square root of the right side: 10.307764064

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.307764064 and -10.307764064.

Subproblem 1

x + 0.5 = 10.307764064 Simplifying x + 0.5 = 10.307764064 Reorder the terms: 0.5 + x = 10.307764064 Solving 0.5 + x = 10.307764064 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 = 10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.307764064 + -0.5 x = 10.307764064 + -0.5 Combine like terms: 10.307764064 + -0.5 = 9.807764064 x = 9.807764064 Simplifying x = 9.807764064

Subproblem 2

x + 0.5 = -10.307764064 Simplifying x + 0.5 = -10.307764064 Reorder the terms: 0.5 + x = -10.307764064 Solving 0.5 + x = -10.307764064 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 = -10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.307764064 + -0.5 x = -10.307764064 + -0.5 Combine like terms: -10.307764064 + -0.5 = -10.807764064 x = -10.807764064 Simplifying x = -10.807764064

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.807764064, -10.807764064}

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