x^2+x=43.5

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Solution for x^2+x=43.5 equation:


Simplifying
x2 + x = 43.5

Reorder the terms:
x + x2 = 43.5

Solving
x + x2 = 43.5

Solving for variable 'x'.

Reorder the terms:
-43.5 + x + x2 = 43.5 + -43.5

Combine like terms: 43.5 + -43.5 = 0.0
-43.5 + x + x2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '43.5' to each side of the equation.
-43.5 + x + 43.5 + x2 = 0.0 + 43.5

Reorder the terms:
-43.5 + 43.5 + x + x2 = 0.0 + 43.5

Combine like terms: -43.5 + 43.5 = 0.0
0.0 + x + x2 = 0.0 + 43.5
x + x2 = 0.0 + 43.5

Combine like terms: 0.0 + 43.5 = 43.5
x + x2 = 43.5

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 = 43.5 + 0.25

Reorder the terms:
0.25 + x + x2 = 43.5 + 0.25

Combine like terms: 43.5 + 0.25 = 43.75
0.25 + x + x2 = 43.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 43.75

Calculate the square root of the right side: 6.614378278

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.614378278 and -6.614378278.

Subproblem 1

x + 0.5 = 6.614378278 Simplifying x + 0.5 = 6.614378278 Reorder the terms: 0.5 + x = 6.614378278 Solving 0.5 + x = 6.614378278 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 = 6.614378278 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.614378278 + -0.5 x = 6.614378278 + -0.5 Combine like terms: 6.614378278 + -0.5 = 6.114378278 x = 6.114378278 Simplifying x = 6.114378278

Subproblem 2

x + 0.5 = -6.614378278 Simplifying x + 0.5 = -6.614378278 Reorder the terms: 0.5 + x = -6.614378278 Solving 0.5 + x = -6.614378278 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 = -6.614378278 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.614378278 + -0.5 x = -6.614378278 + -0.5 Combine like terms: -6.614378278 + -0.5 = -7.114378278 x = -7.114378278 Simplifying x = -7.114378278

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.114378278, -7.114378278}

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