31x^2+21x+3=0

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Solution for 31x^2+21x+3=0 equation:


Simplifying
31x2 + 21x + 3 = 0

Reorder the terms:
3 + 21x + 31x2 = 0

Solving
3 + 21x + 31x2 = 0

Solving for variable 'x'.

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

Divide each side by '31'.
0.09677419355 + 0.6774193548x + x2 = 0

Move the constant term to the right:

Add '-0.09677419355' to each side of the equation.
0.09677419355 + 0.6774193548x + -0.09677419355 + x2 = 0 + -0.09677419355

Reorder the terms:
0.09677419355 + -0.09677419355 + 0.6774193548x + x2 = 0 + -0.09677419355

Combine like terms: 0.09677419355 + -0.09677419355 = 0.00000000000
0.00000000000 + 0.6774193548x + x2 = 0 + -0.09677419355
0.6774193548x + x2 = 0 + -0.09677419355

Combine like terms: 0 + -0.09677419355 = -0.09677419355
0.6774193548x + x2 = -0.09677419355

The x term is 0.6774193548x.  Take half its coefficient (0.3387096774).
Square it (0.1147242456) and add it to both sides.

Add '0.1147242456' to each side of the equation.
0.6774193548x + 0.1147242456 + x2 = -0.09677419355 + 0.1147242456

Reorder the terms:
0.1147242456 + 0.6774193548x + x2 = -0.09677419355 + 0.1147242456

Combine like terms: -0.09677419355 + 0.1147242456 = 0.01795005205
0.1147242456 + 0.6774193548x + x2 = 0.01795005205

Factor a perfect square on the left side:
(x + 0.3387096774)(x + 0.3387096774) = 0.01795005205

Calculate the square root of the right side: 0.133977804

Break this problem into two subproblems by setting 
(x + 0.3387096774) equal to 0.133977804 and -0.133977804.

Subproblem 1

x + 0.3387096774 = 0.133977804 Simplifying x + 0.3387096774 = 0.133977804 Reorder the terms: 0.3387096774 + x = 0.133977804 Solving 0.3387096774 + x = 0.133977804 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3387096774' to each side of the equation. 0.3387096774 + -0.3387096774 + x = 0.133977804 + -0.3387096774 Combine like terms: 0.3387096774 + -0.3387096774 = 0.0000000000 0.0000000000 + x = 0.133977804 + -0.3387096774 x = 0.133977804 + -0.3387096774 Combine like terms: 0.133977804 + -0.3387096774 = -0.2047318734 x = -0.2047318734 Simplifying x = -0.2047318734

Subproblem 2

x + 0.3387096774 = -0.133977804 Simplifying x + 0.3387096774 = -0.133977804 Reorder the terms: 0.3387096774 + x = -0.133977804 Solving 0.3387096774 + x = -0.133977804 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3387096774' to each side of the equation. 0.3387096774 + -0.3387096774 + x = -0.133977804 + -0.3387096774 Combine like terms: 0.3387096774 + -0.3387096774 = 0.0000000000 0.0000000000 + x = -0.133977804 + -0.3387096774 x = -0.133977804 + -0.3387096774 Combine like terms: -0.133977804 + -0.3387096774 = -0.4726874814 x = -0.4726874814 Simplifying x = -0.4726874814

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.2047318734, -0.4726874814}

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