31x*x+21x-85=0

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Solution for 31x*x+21x-85=0 equation:


Simplifying
31x * x + 21x + -85 = 0

Multiply x * x
31x2 + 21x + -85 = 0

Reorder the terms:
-85 + 21x + 31x2 = 0

Solving
-85 + 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'.
-2.741935484 + 0.6774193548x + x2 = 0

Move the constant term to the right:

Add '2.741935484' to each side of the equation.
-2.741935484 + 0.6774193548x + 2.741935484 + x2 = 0 + 2.741935484

Reorder the terms:
-2.741935484 + 2.741935484 + 0.6774193548x + x2 = 0 + 2.741935484

Combine like terms: -2.741935484 + 2.741935484 = 0.000000000
0.000000000 + 0.6774193548x + x2 = 0 + 2.741935484
0.6774193548x + x2 = 0 + 2.741935484

Combine like terms: 0 + 2.741935484 = 2.741935484
0.6774193548x + x2 = 2.741935484

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 = 2.741935484 + 0.1147242456

Reorder the terms:
0.1147242456 + 0.6774193548x + x2 = 2.741935484 + 0.1147242456

Combine like terms: 2.741935484 + 0.1147242456 = 2.8566597296
0.1147242456 + 0.6774193548x + x2 = 2.8566597296

Factor a perfect square on the left side:
(x + 0.3387096774)(x + 0.3387096774) = 2.8566597296

Calculate the square root of the right side: 1.690165592

Break this problem into two subproblems by setting 
(x + 0.3387096774) equal to 1.690165592 and -1.690165592.

Subproblem 1

x + 0.3387096774 = 1.690165592 Simplifying x + 0.3387096774 = 1.690165592 Reorder the terms: 0.3387096774 + x = 1.690165592 Solving 0.3387096774 + x = 1.690165592 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 = 1.690165592 + -0.3387096774 Combine like terms: 0.3387096774 + -0.3387096774 = 0.0000000000 0.0000000000 + x = 1.690165592 + -0.3387096774 x = 1.690165592 + -0.3387096774 Combine like terms: 1.690165592 + -0.3387096774 = 1.3514559146 x = 1.3514559146 Simplifying x = 1.3514559146

Subproblem 2

x + 0.3387096774 = -1.690165592 Simplifying x + 0.3387096774 = -1.690165592 Reorder the terms: 0.3387096774 + x = -1.690165592 Solving 0.3387096774 + x = -1.690165592 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 = -1.690165592 + -0.3387096774 Combine like terms: 0.3387096774 + -0.3387096774 = 0.0000000000 0.0000000000 + x = -1.690165592 + -0.3387096774 x = -1.690165592 + -0.3387096774 Combine like terms: -1.690165592 + -0.3387096774 = -2.0288752694 x = -2.0288752694 Simplifying x = -2.0288752694

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3514559146, -2.0288752694}

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