x^2+x=13

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Solution for x^2+x=13 equation:


Simplifying
x2 + x = 13

Reorder the terms:
x + x2 = 13

Solving
x + x2 = 13

Solving for variable 'x'.

Reorder the terms:
-13 + x + x2 = 13 + -13

Combine like terms: 13 + -13 = 0
-13 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + x + x2 = 0 + 13
x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
x + x2 = 13

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 = 13 + 0.25

Reorder the terms:
0.25 + x + x2 = 13 + 0.25

Combine like terms: 13 + 0.25 = 13.25
0.25 + x + x2 = 13.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

x + 0.5 = 3.640054945 Simplifying x + 0.5 = 3.640054945 Reorder the terms: 0.5 + x = 3.640054945 Solving 0.5 + x = 3.640054945 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 = 3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.640054945 + -0.5 x = 3.640054945 + -0.5 Combine like terms: 3.640054945 + -0.5 = 3.140054945 x = 3.140054945 Simplifying x = 3.140054945

Subproblem 2

x + 0.5 = -3.640054945 Simplifying x + 0.5 = -3.640054945 Reorder the terms: 0.5 + x = -3.640054945 Solving 0.5 + x = -3.640054945 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 = -3.640054945 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.640054945 + -0.5 x = -3.640054945 + -0.5 Combine like terms: -3.640054945 + -0.5 = -4.140054945 x = -4.140054945 Simplifying x = -4.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.140054945, -4.140054945}

 

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