3x^2+13x-31=0

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Solution for 3x^2+13x-31=0 equation:


Simplifying
3x2 + 13x + -31 = 0

Reorder the terms:
-31 + 13x + 3x2 = 0

Solving
-31 + 13x + 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'.
-10.33333333 + 4.333333333x + x2 = 0

Move the constant term to the right:

Add '10.33333333' to each side of the equation.
-10.33333333 + 4.333333333x + 10.33333333 + x2 = 0 + 10.33333333

Reorder the terms:
-10.33333333 + 10.33333333 + 4.333333333x + x2 = 0 + 10.33333333

Combine like terms: -10.33333333 + 10.33333333 = 0.00000000
0.00000000 + 4.333333333x + x2 = 0 + 10.33333333
4.333333333x + x2 = 0 + 10.33333333

Combine like terms: 0 + 10.33333333 = 10.33333333
4.333333333x + x2 = 10.33333333

The x term is 4.333333333x.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333x + 4.694444446 + x2 = 10.33333333 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333x + x2 = 10.33333333 + 4.694444446

Combine like terms: 10.33333333 + 4.694444446 = 15.027777776
4.694444446 + 4.333333333x + x2 = 15.027777776

Factor a perfect square on the left side:
(x + 2.166666667)(x + 2.166666667) = 15.027777776

Calculate the square root of the right side: 3.876567783

Break this problem into two subproblems by setting 
(x + 2.166666667) equal to 3.876567783 and -3.876567783.

Subproblem 1

x + 2.166666667 = 3.876567783 Simplifying x + 2.166666667 = 3.876567783 Reorder the terms: 2.166666667 + x = 3.876567783 Solving 2.166666667 + x = 3.876567783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + x = 3.876567783 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + x = 3.876567783 + -2.166666667 x = 3.876567783 + -2.166666667 Combine like terms: 3.876567783 + -2.166666667 = 1.709901116 x = 1.709901116 Simplifying x = 1.709901116

Subproblem 2

x + 2.166666667 = -3.876567783 Simplifying x + 2.166666667 = -3.876567783 Reorder the terms: 2.166666667 + x = -3.876567783 Solving 2.166666667 + x = -3.876567783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + x = -3.876567783 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + x = -3.876567783 + -2.166666667 x = -3.876567783 + -2.166666667 Combine like terms: -3.876567783 + -2.166666667 = -6.04323445 x = -6.04323445 Simplifying x = -6.04323445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.709901116, -6.04323445}

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