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