x^2+x=68

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Solution for x^2+x=68 equation:


Simplifying
x2 + x = 68

Reorder the terms:
x + x2 = 68

Solving
x + x2 = 68

Solving for variable 'x'.

Reorder the terms:
-68 + x + x2 = 68 + -68

Combine like terms: 68 + -68 = 0
-68 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '68' to each side of the equation.
-68 + x + 68 + x2 = 0 + 68

Reorder the terms:
-68 + 68 + x + x2 = 0 + 68

Combine like terms: -68 + 68 = 0
0 + x + x2 = 0 + 68
x + x2 = 0 + 68

Combine like terms: 0 + 68 = 68
x + x2 = 68

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 = 68 + 0.25

Reorder the terms:
0.25 + x + x2 = 68 + 0.25

Combine like terms: 68 + 0.25 = 68.25
0.25 + x + x2 = 68.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 68.25

Calculate the square root of the right side: 8.261355821

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.261355821 and -8.261355821.

Subproblem 1

x + 0.5 = 8.261355821 Simplifying x + 0.5 = 8.261355821 Reorder the terms: 0.5 + x = 8.261355821 Solving 0.5 + x = 8.261355821 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 = 8.261355821 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.261355821 + -0.5 x = 8.261355821 + -0.5 Combine like terms: 8.261355821 + -0.5 = 7.761355821 x = 7.761355821 Simplifying x = 7.761355821

Subproblem 2

x + 0.5 = -8.261355821 Simplifying x + 0.5 = -8.261355821 Reorder the terms: 0.5 + x = -8.261355821 Solving 0.5 + x = -8.261355821 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 = -8.261355821 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.261355821 + -0.5 x = -8.261355821 + -0.5 Combine like terms: -8.261355821 + -0.5 = -8.761355821 x = -8.761355821 Simplifying x = -8.761355821

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.761355821, -8.761355821}

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