x^2+x=67

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Solution for x^2+x=67 equation:


Simplifying
x2 + x = 67

Reorder the terms:
x + x2 = 67

Solving
x + x2 = 67

Solving for variable 'x'.

Reorder the terms:
-67 + x + x2 = 67 + -67

Combine like terms: 67 + -67 = 0
-67 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '67' to each side of the equation.
-67 + x + 67 + x2 = 0 + 67

Reorder the terms:
-67 + 67 + x + x2 = 0 + 67

Combine like terms: -67 + 67 = 0
0 + x + x2 = 0 + 67
x + x2 = 0 + 67

Combine like terms: 0 + 67 = 67
x + x2 = 67

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 = 67 + 0.25

Reorder the terms:
0.25 + x + x2 = 67 + 0.25

Combine like terms: 67 + 0.25 = 67.25
0.25 + x + x2 = 67.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 67.25

Calculate the square root of the right side: 8.200609733

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.200609733 and -8.200609733.

Subproblem 1

x + 0.5 = 8.200609733 Simplifying x + 0.5 = 8.200609733 Reorder the terms: 0.5 + x = 8.200609733 Solving 0.5 + x = 8.200609733 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 = 8.200609733 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.200609733 + -0.5 x = 8.200609733 + -0.5 Combine like terms: 8.200609733 + -0.5 = 7.700609733 x = 7.700609733 Simplifying x = 7.700609733

Subproblem 2

x + 0.5 = -8.200609733 Simplifying x + 0.5 = -8.200609733 Reorder the terms: 0.5 + x = -8.200609733 Solving 0.5 + x = -8.200609733 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 = -8.200609733 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.200609733 + -0.5 x = -8.200609733 + -0.5 Combine like terms: -8.200609733 + -0.5 = -8.700609733 x = -8.700609733 Simplifying x = -8.700609733

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.700609733, -8.700609733}

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