3*x^2-31*x+5=0

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Solution for 3*x^2-31*x+5=0 equation:


Simplifying
3x2 + -31x + 5 = 0

Reorder the terms:
5 + -31x + 3x2 = 0

Solving
5 + -31x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
1.666666667 + -10.33333333x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -10.33333333x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -10.33333333x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -10.33333333x + x2 = 0 + -1.666666667
-10.33333333x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-10.33333333x + x2 = -1.666666667

The x term is -10.33333333x.  Take half its coefficient (-5.166666665).
Square it (26.69444443) and add it to both sides.

Add '26.69444443' to each side of the equation.
-10.33333333x + 26.69444443 + x2 = -1.666666667 + 26.69444443

Reorder the terms:
26.69444443 + -10.33333333x + x2 = -1.666666667 + 26.69444443

Combine like terms: -1.666666667 + 26.69444443 = 25.027777763
26.69444443 + -10.33333333x + x2 = 25.027777763

Factor a perfect square on the left side:
(x + -5.166666665)(x + -5.166666665) = 25.027777763

Calculate the square root of the right side: 5.002777005

Break this problem into two subproblems by setting 
(x + -5.166666665) equal to 5.002777005 and -5.002777005.

Subproblem 1

x + -5.166666665 = 5.002777005 Simplifying x + -5.166666665 = 5.002777005 Reorder the terms: -5.166666665 + x = 5.002777005 Solving -5.166666665 + x = 5.002777005 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.166666665' to each side of the equation. -5.166666665 + 5.166666665 + x = 5.002777005 + 5.166666665 Combine like terms: -5.166666665 + 5.166666665 = 0.000000000 0.000000000 + x = 5.002777005 + 5.166666665 x = 5.002777005 + 5.166666665 Combine like terms: 5.002777005 + 5.166666665 = 10.16944367 x = 10.16944367 Simplifying x = 10.16944367

Subproblem 2

x + -5.166666665 = -5.002777005 Simplifying x + -5.166666665 = -5.002777005 Reorder the terms: -5.166666665 + x = -5.002777005 Solving -5.166666665 + x = -5.002777005 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.166666665' to each side of the equation. -5.166666665 + 5.166666665 + x = -5.002777005 + 5.166666665 Combine like terms: -5.166666665 + 5.166666665 = 0.000000000 0.000000000 + x = -5.002777005 + 5.166666665 x = -5.002777005 + 5.166666665 Combine like terms: -5.002777005 + 5.166666665 = 0.16388966 x = 0.16388966 Simplifying x = 0.16388966

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.16944367, 0.16388966}

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