x+x^2=84

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Solution for x+x^2=84 equation:


Simplifying
x + x2 = 84

Solving
x + x2 = 84

Solving for variable 'x'.

Reorder the terms:
-84 + x + x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + x + x2 = 0 + 84
x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
x + x2 = 84

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 = 84 + 0.25

Reorder the terms:
0.25 + x + x2 = 84 + 0.25

Combine like terms: 84 + 0.25 = 84.25
0.25 + x + x2 = 84.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(x + 0.5) equal to 9.178779875 and -9.178779875.

Subproblem 1

x + 0.5 = 9.178779875 Simplifying x + 0.5 = 9.178779875 Reorder the terms: 0.5 + x = 9.178779875 Solving 0.5 + x = 9.178779875 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 = 9.178779875 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 9.178779875 + -0.5 x = 9.178779875 + -0.5 Combine like terms: 9.178779875 + -0.5 = 8.678779875 x = 8.678779875 Simplifying x = 8.678779875

Subproblem 2

x + 0.5 = -9.178779875 Simplifying x + 0.5 = -9.178779875 Reorder the terms: 0.5 + x = -9.178779875 Solving 0.5 + x = -9.178779875 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 = -9.178779875 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -9.178779875 + -0.5 x = -9.178779875 + -0.5 Combine like terms: -9.178779875 + -0.5 = -9.678779875 x = -9.678779875 Simplifying x = -9.678779875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.678779875, -9.678779875}

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