x^2+x=840

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Solution for x^2+x=840 equation:


Simplifying
x2 + x = 840

Reorder the terms:
x + x2 = 840

Solving
x + x2 = 840

Solving for variable 'x'.

Reorder the terms:
-840 + x + x2 = 840 + -840

Combine like terms: 840 + -840 = 0
-840 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '840' to each side of the equation.
-840 + x + 840 + x2 = 0 + 840

Reorder the terms:
-840 + 840 + x + x2 = 0 + 840

Combine like terms: -840 + 840 = 0
0 + x + x2 = 0 + 840
x + x2 = 0 + 840

Combine like terms: 0 + 840 = 840
x + x2 = 840

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 = 840 + 0.25

Reorder the terms:
0.25 + x + x2 = 840 + 0.25

Combine like terms: 840 + 0.25 = 840.25
0.25 + x + x2 = 840.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 840.25

Calculate the square root of the right side: 28.987066081

Break this problem into two subproblems by setting 
(x + 0.5) equal to 28.987066081 and -28.987066081.

Subproblem 1

x + 0.5 = 28.987066081 Simplifying x + 0.5 = 28.987066081 Reorder the terms: 0.5 + x = 28.987066081 Solving 0.5 + x = 28.987066081 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 = 28.987066081 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 28.987066081 + -0.5 x = 28.987066081 + -0.5 Combine like terms: 28.987066081 + -0.5 = 28.487066081 x = 28.487066081 Simplifying x = 28.487066081

Subproblem 2

x + 0.5 = -28.987066081 Simplifying x + 0.5 = -28.987066081 Reorder the terms: 0.5 + x = -28.987066081 Solving 0.5 + x = -28.987066081 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 = -28.987066081 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -28.987066081 + -0.5 x = -28.987066081 + -0.5 Combine like terms: -28.987066081 + -0.5 = -29.487066081 x = -29.487066081 Simplifying x = -29.487066081

Solution

The solution to the problem is based on the solutions from the subproblems. x = {28.487066081, -29.487066081}

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