x^2+x=35678

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Solution for x^2+x=35678 equation:


Simplifying
x2 + x = 35678

Reorder the terms:
x + x2 = 35678

Solving
x + x2 = 35678

Solving for variable 'x'.

Reorder the terms:
-35678 + x + x2 = 35678 + -35678

Combine like terms: 35678 + -35678 = 0
-35678 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '35678' to each side of the equation.
-35678 + x + 35678 + x2 = 0 + 35678

Reorder the terms:
-35678 + 35678 + x + x2 = 0 + 35678

Combine like terms: -35678 + 35678 = 0
0 + x + x2 = 0 + 35678
x + x2 = 0 + 35678

Combine like terms: 0 + 35678 = 35678
x + x2 = 35678

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 = 35678 + 0.25

Reorder the terms:
0.25 + x + x2 = 35678 + 0.25

Combine like terms: 35678 + 0.25 = 35678.25
0.25 + x + x2 = 35678.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 35678.25

Calculate the square root of the right side: 188.886870904

Break this problem into two subproblems by setting 
(x + 0.5) equal to 188.886870904 and -188.886870904.

Subproblem 1

x + 0.5 = 188.886870904 Simplifying x + 0.5 = 188.886870904 Reorder the terms: 0.5 + x = 188.886870904 Solving 0.5 + x = 188.886870904 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 = 188.886870904 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 188.886870904 + -0.5 x = 188.886870904 + -0.5 Combine like terms: 188.886870904 + -0.5 = 188.386870904 x = 188.386870904 Simplifying x = 188.386870904

Subproblem 2

x + 0.5 = -188.886870904 Simplifying x + 0.5 = -188.886870904 Reorder the terms: 0.5 + x = -188.886870904 Solving 0.5 + x = -188.886870904 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 = -188.886870904 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -188.886870904 + -0.5 x = -188.886870904 + -0.5 Combine like terms: -188.886870904 + -0.5 = -189.386870904 x = -189.386870904 Simplifying x = -189.386870904

Solution

The solution to the problem is based on the solutions from the subproblems. x = {188.386870904, -189.386870904}

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