0=x^2-10x-31

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Solution for 0=x^2-10x-31 equation:


Simplifying
0 = x2 + -10x + -31

Reorder the terms:
0 = -31 + -10x + x2

Solving
0 = -31 + -10x + x2

Solving for variable 'x'.

Combine like terms: 0 + 31 = 31
31 + 10x + -1x2 = -31 + -10x + x2 + 31 + 10x + -1x2

Reorder the terms:
31 + 10x + -1x2 = -31 + 31 + -10x + 10x + x2 + -1x2

Combine like terms: -31 + 31 = 0
31 + 10x + -1x2 = 0 + -10x + 10x + x2 + -1x2
31 + 10x + -1x2 = -10x + 10x + x2 + -1x2

Combine like terms: -10x + 10x = 0
31 + 10x + -1x2 = 0 + x2 + -1x2
31 + 10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
31 + 10x + -1x2 = 0

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

Divide each side by '-1'.
-31 + -10x + x2 = 0

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -10x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + -10x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -10x + x2 = 0 + 31
-10x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
-10x + x2 = 31

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = 31 + 25

Reorder the terms:
25 + -10x + x2 = 31 + 25

Combine like terms: 31 + 25 = 56
25 + -10x + x2 = 56

Factor a perfect square on the left side:
(x + -5)(x + -5) = 56

Calculate the square root of the right side: 7.483314774

Break this problem into two subproblems by setting 
(x + -5) equal to 7.483314774 and -7.483314774.

Subproblem 1

x + -5 = 7.483314774 Simplifying x + -5 = 7.483314774 Reorder the terms: -5 + x = 7.483314774 Solving -5 + x = 7.483314774 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 7.483314774 + 5 Combine like terms: -5 + 5 = 0 0 + x = 7.483314774 + 5 x = 7.483314774 + 5 Combine like terms: 7.483314774 + 5 = 12.483314774 x = 12.483314774 Simplifying x = 12.483314774

Subproblem 2

x + -5 = -7.483314774 Simplifying x + -5 = -7.483314774 Reorder the terms: -5 + x = -7.483314774 Solving -5 + x = -7.483314774 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -7.483314774 + 5 Combine like terms: -5 + 5 = 0 0 + x = -7.483314774 + 5 x = -7.483314774 + 5 Combine like terms: -7.483314774 + 5 = -2.483314774 x = -2.483314774 Simplifying x = -2.483314774

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.483314774, -2.483314774}

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