0=x^2+x-189

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Solution for 0=x^2+x-189 equation:


Simplifying
0 = x2 + x + -189

Reorder the terms:
0 = -189 + x + x2

Solving
0 = -189 + x + x2

Solving for variable 'x'.

Combine like terms: 0 + 189 = 189
189 + -1x + -1x2 = -189 + x + x2 + 189 + -1x + -1x2

Reorder the terms:
189 + -1x + -1x2 = -189 + 189 + x + -1x + x2 + -1x2

Combine like terms: -189 + 189 = 0
189 + -1x + -1x2 = 0 + x + -1x + x2 + -1x2
189 + -1x + -1x2 = x + -1x + x2 + -1x2

Combine like terms: x + -1x = 0
189 + -1x + -1x2 = 0 + x2 + -1x2
189 + -1x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
189 + -1x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-189 + x + x2 = 0

Move the constant term to the right:

Add '189' to each side of the equation.
-189 + x + 189 + x2 = 0 + 189

Reorder the terms:
-189 + 189 + x + x2 = 0 + 189

Combine like terms: -189 + 189 = 0
0 + x + x2 = 0 + 189
x + x2 = 0 + 189

Combine like terms: 0 + 189 = 189
x + x2 = 189

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.
 + 0.25 + x2 = 189 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 189 + 0.25

Combine like terms: 189 + 0.25 = 189.25
1.25 + x2 = 189.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 189.25

Calculate the square root of the right side: 13.756816492

Break this problem into two subproblems by setting 
(x + 0.5) equal to 13.756816492 and -13.756816492.

Subproblem 1

x + 0.5 = 13.756816492 Simplifying x + 0.5 = 13.756816492 Reorder the terms: 0.5 + x = 13.756816492 Solving 0.5 + x = 13.756816492 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 = 13.756816492 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 13.756816492 + -0.5 x = 13.756816492 + -0.5 Combine like terms: 13.756816492 + -0.5 = 13.256816492 x = 13.256816492 Simplifying x = 13.256816492

Subproblem 2

x + 0.5 = -13.756816492 Simplifying x + 0.5 = -13.756816492 Reorder the terms: 0.5 + x = -13.756816492 Solving 0.5 + x = -13.756816492 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 = -13.756816492 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -13.756816492 + -0.5 x = -13.756816492 + -0.5 Combine like terms: -13.756816492 + -0.5 = -14.256816492 x = -14.256816492 Simplifying x = -14.256816492

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.256816492, -14.256816492}

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