3x^2+31x-420=0

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Solution for 3x^2+31x-420=0 equation:


Simplifying
3x2 + 31x + -420 = 0

Reorder the terms:
-420 + 31x + 3x2 = 0

Solving
-420 + 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'.
-140 + 10.33333333x + x2 = 0

Move the constant term to the right:

Add '140' to each side of the equation.
-140 + 10.33333333x + 140 + x2 = 0 + 140

Reorder the terms:
-140 + 140 + 10.33333333x + x2 = 0 + 140

Combine like terms: -140 + 140 = 0
0 + 10.33333333x + x2 = 0 + 140
10.33333333x + x2 = 0 + 140

Combine like terms: 0 + 140 = 140
10.33333333x + x2 = 140

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 = 140 + 26.69444443

Reorder the terms:
26.69444443 + 10.33333333x + x2 = 140 + 26.69444443

Combine like terms: 140 + 26.69444443 = 166.69444443
26.69444443 + 10.33333333x + x2 = 166.69444443

Factor a perfect square on the left side:
(x + 5.166666665)(x + 5.166666665) = 166.69444443

Calculate the square root of the right side: 12.911020271

Break this problem into two subproblems by setting 
(x + 5.166666665) equal to 12.911020271 and -12.911020271.

Subproblem 1

x + 5.166666665 = 12.911020271 Simplifying x + 5.166666665 = 12.911020271 Reorder the terms: 5.166666665 + x = 12.911020271 Solving 5.166666665 + x = 12.911020271 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 = 12.911020271 + -5.166666665 Combine like terms: 5.166666665 + -5.166666665 = 0.000000000 0.000000000 + x = 12.911020271 + -5.166666665 x = 12.911020271 + -5.166666665 Combine like terms: 12.911020271 + -5.166666665 = 7.744353606 x = 7.744353606 Simplifying x = 7.744353606

Subproblem 2

x + 5.166666665 = -12.911020271 Simplifying x + 5.166666665 = -12.911020271 Reorder the terms: 5.166666665 + x = -12.911020271 Solving 5.166666665 + x = -12.911020271 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 = -12.911020271 + -5.166666665 Combine like terms: 5.166666665 + -5.166666665 = 0.000000000 0.000000000 + x = -12.911020271 + -5.166666665 x = -12.911020271 + -5.166666665 Combine like terms: -12.911020271 + -5.166666665 = -18.077686936 x = -18.077686936 Simplifying x = -18.077686936

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.744353606, -18.077686936}

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