x^2+x=31

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Solution for x^2+x=31 equation:


Simplifying
x2 + x = 31

Reorder the terms:
x + x2 = 31

Solving
x + x2 = 31

Solving for variable 'x'.

Reorder the terms:
-31 + x + x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + x + x2 = 0 + 31
x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
x + x2 = 31

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 = 31 + 0.25

Reorder the terms:
0.25 + x + x2 = 31 + 0.25

Combine like terms: 31 + 0.25 = 31.25
0.25 + x + x2 = 31.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 31.25

Calculate the square root of the right side: 5.590169944

Break this problem into two subproblems by setting 
(x + 0.5) equal to 5.590169944 and -5.590169944.

Subproblem 1

x + 0.5 = 5.590169944 Simplifying x + 0.5 = 5.590169944 Reorder the terms: 0.5 + x = 5.590169944 Solving 0.5 + x = 5.590169944 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 = 5.590169944 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 5.590169944 + -0.5 x = 5.590169944 + -0.5 Combine like terms: 5.590169944 + -0.5 = 5.090169944 x = 5.090169944 Simplifying x = 5.090169944

Subproblem 2

x + 0.5 = -5.590169944 Simplifying x + 0.5 = -5.590169944 Reorder the terms: 0.5 + x = -5.590169944 Solving 0.5 + x = -5.590169944 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 = -5.590169944 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -5.590169944 + -0.5 x = -5.590169944 + -0.5 Combine like terms: -5.590169944 + -0.5 = -6.090169944 x = -6.090169944 Simplifying x = -6.090169944

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.090169944, -6.090169944}

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