x^2+x=308

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Solution for x^2+x=308 equation:


Simplifying
x2 + x = 308

Reorder the terms:
x + x2 = 308

Solving
x + x2 = 308

Solving for variable 'x'.

Reorder the terms:
-308 + x + x2 = 308 + -308

Combine like terms: 308 + -308 = 0
-308 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '308' to each side of the equation.
-308 + x + 308 + x2 = 0 + 308

Reorder the terms:
-308 + 308 + x + x2 = 0 + 308

Combine like terms: -308 + 308 = 0
0 + x + x2 = 0 + 308
x + x2 = 0 + 308

Combine like terms: 0 + 308 = 308
x + x2 = 308

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 = 308 + 0.25

Reorder the terms:
0.25 + x + x2 = 308 + 0.25

Combine like terms: 308 + 0.25 = 308.25
0.25 + x + x2 = 308.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 308.25

Calculate the square root of the right side: 17.557049866

Break this problem into two subproblems by setting 
(x + 0.5) equal to 17.557049866 and -17.557049866.

Subproblem 1

x + 0.5 = 17.557049866 Simplifying x + 0.5 = 17.557049866 Reorder the terms: 0.5 + x = 17.557049866 Solving 0.5 + x = 17.557049866 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 = 17.557049866 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 17.557049866 + -0.5 x = 17.557049866 + -0.5 Combine like terms: 17.557049866 + -0.5 = 17.057049866 x = 17.057049866 Simplifying x = 17.057049866

Subproblem 2

x + 0.5 = -17.557049866 Simplifying x + 0.5 = -17.557049866 Reorder the terms: 0.5 + x = -17.557049866 Solving 0.5 + x = -17.557049866 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 = -17.557049866 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -17.557049866 + -0.5 x = -17.557049866 + -0.5 Combine like terms: -17.557049866 + -0.5 = -18.057049866 x = -18.057049866 Simplifying x = -18.057049866

Solution

The solution to the problem is based on the solutions from the subproblems. x = {17.057049866, -18.057049866}

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