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Simplifying 81x2 + -310x + -2000 = 0 Reorder the terms: -2000 + -310x + 81x2 = 0 Solving -2000 + -310x + 81x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -24.69135802 + -3.827160494x + x2 = 0 Move the constant term to the right: Add '24.69135802' to each side of the equation. -24.69135802 + -3.827160494x + 24.69135802 + x2 = 0 + 24.69135802 Reorder the terms: -24.69135802 + 24.69135802 + -3.827160494x + x2 = 0 + 24.69135802 Combine like terms: -24.69135802 + 24.69135802 = 0.00000000 0.00000000 + -3.827160494x + x2 = 0 + 24.69135802 -3.827160494x + x2 = 0 + 24.69135802 Combine like terms: 0 + 24.69135802 = 24.69135802 -3.827160494x + x2 = 24.69135802 The x term is -3.827160494x. Take half its coefficient (-1.913580247). Square it (3.661789362) and add it to both sides. Add '3.661789362' to each side of the equation. -3.827160494x + 3.661789362 + x2 = 24.69135802 + 3.661789362 Reorder the terms: 3.661789362 + -3.827160494x + x2 = 24.69135802 + 3.661789362 Combine like terms: 24.69135802 + 3.661789362 = 28.353147382 3.661789362 + -3.827160494x + x2 = 28.353147382 Factor a perfect square on the left side: (x + -1.913580247)(x + -1.913580247) = 28.353147382 Calculate the square root of the right side: 5.324767355 Break this problem into two subproblems by setting (x + -1.913580247) equal to 5.324767355 and -5.324767355.Subproblem 1
x + -1.913580247 = 5.324767355 Simplifying x + -1.913580247 = 5.324767355 Reorder the terms: -1.913580247 + x = 5.324767355 Solving -1.913580247 + x = 5.324767355 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.913580247' to each side of the equation. -1.913580247 + 1.913580247 + x = 5.324767355 + 1.913580247 Combine like terms: -1.913580247 + 1.913580247 = 0.000000000 0.000000000 + x = 5.324767355 + 1.913580247 x = 5.324767355 + 1.913580247 Combine like terms: 5.324767355 + 1.913580247 = 7.238347602 x = 7.238347602 Simplifying x = 7.238347602Subproblem 2
x + -1.913580247 = -5.324767355 Simplifying x + -1.913580247 = -5.324767355 Reorder the terms: -1.913580247 + x = -5.324767355 Solving -1.913580247 + x = -5.324767355 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.913580247' to each side of the equation. -1.913580247 + 1.913580247 + x = -5.324767355 + 1.913580247 Combine like terms: -1.913580247 + 1.913580247 = 0.000000000 0.000000000 + x = -5.324767355 + 1.913580247 x = -5.324767355 + 1.913580247 Combine like terms: -5.324767355 + 1.913580247 = -3.411187108 x = -3.411187108 Simplifying x = -3.411187108Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.238347602, -3.411187108}
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