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Simplifying 0 = 27x2 + 8x + -91 Reorder the terms: 0 = -91 + 8x + 27x2 Solving 0 = -91 + 8x + 27x2 Solving for variable 'x'. Combine like terms: 0 + 91 = 91 91 + -8x + -27x2 = -91 + 8x + 27x2 + 91 + -8x + -27x2 Reorder the terms: 91 + -8x + -27x2 = -91 + 91 + 8x + -8x + 27x2 + -27x2 Combine like terms: -91 + 91 = 0 91 + -8x + -27x2 = 0 + 8x + -8x + 27x2 + -27x2 91 + -8x + -27x2 = 8x + -8x + 27x2 + -27x2 Combine like terms: 8x + -8x = 0 91 + -8x + -27x2 = 0 + 27x2 + -27x2 91 + -8x + -27x2 = 27x2 + -27x2 Combine like terms: 27x2 + -27x2 = 0 91 + -8x + -27x2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -3.37037037 + 0.2962962963x + x2 = 0 Move the constant term to the right: Add '3.37037037' to each side of the equation. -3.37037037 + 0.2962962963x + 3.37037037 + x2 = 0 + 3.37037037 Reorder the terms: -3.37037037 + 3.37037037 + 0.2962962963x + x2 = 0 + 3.37037037 Combine like terms: -3.37037037 + 3.37037037 = 0.00000000 0.00000000 + 0.2962962963x + x2 = 0 + 3.37037037 0.2962962963x + x2 = 0 + 3.37037037 Combine like terms: 0 + 3.37037037 = 3.37037037 0.2962962963x + x2 = 3.37037037 The x term is 0.2962962963x. Take half its coefficient (0.1481481482). Square it (0.02194787382) and add it to both sides. Add '0.02194787382' to each side of the equation. 0.2962962963x + 0.02194787382 + x2 = 3.37037037 + 0.02194787382 Reorder the terms: 0.02194787382 + 0.2962962963x + x2 = 3.37037037 + 0.02194787382 Combine like terms: 3.37037037 + 0.02194787382 = 3.39231824382 0.02194787382 + 0.2962962963x + x2 = 3.39231824382 Factor a perfect square on the left side: (x + 0.1481481482)(x + 0.1481481482) = 3.39231824382 Calculate the square root of the right side: 1.841824705 Break this problem into two subproblems by setting (x + 0.1481481482) equal to 1.841824705 and -1.841824705.Subproblem 1
x + 0.1481481482 = 1.841824705 Simplifying x + 0.1481481482 = 1.841824705 Reorder the terms: 0.1481481482 + x = 1.841824705 Solving 0.1481481482 + x = 1.841824705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1481481482' to each side of the equation. 0.1481481482 + -0.1481481482 + x = 1.841824705 + -0.1481481482 Combine like terms: 0.1481481482 + -0.1481481482 = 0.0000000000 0.0000000000 + x = 1.841824705 + -0.1481481482 x = 1.841824705 + -0.1481481482 Combine like terms: 1.841824705 + -0.1481481482 = 1.6936765568 x = 1.6936765568 Simplifying x = 1.6936765568Subproblem 2
x + 0.1481481482 = -1.841824705 Simplifying x + 0.1481481482 = -1.841824705 Reorder the terms: 0.1481481482 + x = -1.841824705 Solving 0.1481481482 + x = -1.841824705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1481481482' to each side of the equation. 0.1481481482 + -0.1481481482 + x = -1.841824705 + -0.1481481482 Combine like terms: 0.1481481482 + -0.1481481482 = 0.0000000000 0.0000000000 + x = -1.841824705 + -0.1481481482 x = -1.841824705 + -0.1481481482 Combine like terms: -1.841824705 + -0.1481481482 = -1.9899728532 x = -1.9899728532 Simplifying x = -1.9899728532Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.6936765568, -1.9899728532}
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