0=x^2-x-40

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Solution for 0=x^2-x-40 equation:


Simplifying
0 = x2 + -1x + -40

Reorder the terms:
0 = -40 + -1x + x2

Solving
0 = -40 + -1x + x2

Solving for variable 'x'.

Combine like terms: 0 + 40 = 40
40 + x + -1x2 = -40 + -1x + x2 + 40 + x + -1x2

Reorder the terms:
40 + x + -1x2 = -40 + 40 + -1x + x + x2 + -1x2

Combine like terms: -40 + 40 = 0
40 + x + -1x2 = 0 + -1x + x + x2 + -1x2
40 + x + -1x2 = -1x + x + x2 + -1x2

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

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

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

Divide each side by '-1'.
-40 + -1x + x2 = 0

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + -1x + 40 + x2 = 0 + 40

Reorder the terms:
-40 + 40 + -1x + x2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + -1x + x2 = 0 + 40
-1x + x2 = 0 + 40

Combine like terms: 0 + 40 = 40
-1x + x2 = 40

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.
-1x + 0.25 + x2 = 40 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 40 + 0.25

Combine like terms: 40 + 0.25 = 40.25
0.25 + -1x + x2 = 40.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.34428877 and -6.34428877.

Subproblem 1

x + 0.5 = 6.34428877 Simplifying x + 0.5 = 6.34428877 Reorder the terms: 0.5 + x = 6.34428877 Solving 0.5 + x = 6.34428877 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 = 6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.34428877 + -0.5 x = 6.34428877 + -0.5 Combine like terms: 6.34428877 + -0.5 = 5.84428877 x = 5.84428877 Simplifying x = 5.84428877

Subproblem 2

x + 0.5 = -6.34428877 Simplifying x + 0.5 = -6.34428877 Reorder the terms: 0.5 + x = -6.34428877 Solving 0.5 + x = -6.34428877 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 = -6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.34428877 + -0.5 x = -6.34428877 + -0.5 Combine like terms: -6.34428877 + -0.5 = -6.84428877 x = -6.84428877 Simplifying x = -6.84428877

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.84428877, -6.84428877}

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