6x^2+31x=21

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Solution for 6x^2+31x=21 equation:


Simplifying
6x2 + 31x = 21

Reorder the terms:
31x + 6x2 = 21

Solving
31x + 6x2 = 21

Solving for variable 'x'.

Reorder the terms:
-21 + 31x + 6x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 31x + 6x2 = 0

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

Divide each side by '6'.
-3.5 + 5.166666667x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 5.166666667x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 5.166666667x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 5.166666667x + x2 = 0 + 3.5
5.166666667x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
5.166666667x + x2 = 3.5

The x term is 5.166666667x.  Take half its coefficient (2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
5.166666667x + 6.673611115 + x2 = 3.5 + 6.673611115

Reorder the terms:
6.673611115 + 5.166666667x + x2 = 3.5 + 6.673611115

Combine like terms: 3.5 + 6.673611115 = 10.173611115
6.673611115 + 5.166666667x + x2 = 10.173611115

Factor a perfect square on the left side:
(x + 2.583333334)(x + 2.583333334) = 10.173611115

Calculate the square root of the right side: 3.189609869

Break this problem into two subproblems by setting 
(x + 2.583333334) equal to 3.189609869 and -3.189609869.

Subproblem 1

x + 2.583333334 = 3.189609869 Simplifying x + 2.583333334 = 3.189609869 Reorder the terms: 2.583333334 + x = 3.189609869 Solving 2.583333334 + x = 3.189609869 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + x = 3.189609869 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + x = 3.189609869 + -2.583333334 x = 3.189609869 + -2.583333334 Combine like terms: 3.189609869 + -2.583333334 = 0.606276535 x = 0.606276535 Simplifying x = 0.606276535

Subproblem 2

x + 2.583333334 = -3.189609869 Simplifying x + 2.583333334 = -3.189609869 Reorder the terms: 2.583333334 + x = -3.189609869 Solving 2.583333334 + x = -3.189609869 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.583333334' to each side of the equation. 2.583333334 + -2.583333334 + x = -3.189609869 + -2.583333334 Combine like terms: 2.583333334 + -2.583333334 = 0.000000000 0.000000000 + x = -3.189609869 + -2.583333334 x = -3.189609869 + -2.583333334 Combine like terms: -3.189609869 + -2.583333334 = -5.772943203 x = -5.772943203 Simplifying x = -5.772943203

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.606276535, -5.772943203}

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