x*x+x=65

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Solution for x*x+x=65 equation:


Simplifying
x * x + x = 65

Multiply x * x
x2 + x = 65

Reorder the terms:
x + x2 = 65

Solving
x + x2 = 65

Solving for variable 'x'.

Reorder the terms:
-65 + x + x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-65 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '65' to each side of the equation.
-65 + x + 65 + x2 = 0 + 65

Reorder the terms:
-65 + 65 + x + x2 = 0 + 65

Combine like terms: -65 + 65 = 0
0 + x + x2 = 0 + 65
x + x2 = 0 + 65

Combine like terms: 0 + 65 = 65
x + x2 = 65

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 = 65 + 0.25

Reorder the terms:
0.25 + x + x2 = 65 + 0.25

Combine like terms: 65 + 0.25 = 65.25
0.25 + x + x2 = 65.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 65.25

Calculate the square root of the right side: 8.077747211

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.077747211 and -8.077747211.

Subproblem 1

x + 0.5 = 8.077747211 Simplifying x + 0.5 = 8.077747211 Reorder the terms: 0.5 + x = 8.077747211 Solving 0.5 + x = 8.077747211 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 = 8.077747211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.077747211 + -0.5 x = 8.077747211 + -0.5 Combine like terms: 8.077747211 + -0.5 = 7.577747211 x = 7.577747211 Simplifying x = 7.577747211

Subproblem 2

x + 0.5 = -8.077747211 Simplifying x + 0.5 = -8.077747211 Reorder the terms: 0.5 + x = -8.077747211 Solving 0.5 + x = -8.077747211 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 = -8.077747211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.077747211 + -0.5 x = -8.077747211 + -0.5 Combine like terms: -8.077747211 + -0.5 = -8.577747211 x = -8.577747211 Simplifying x = -8.577747211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.577747211, -8.577747211}

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