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Simplifying 1077 = 27x2 + 40x Reorder the terms: 1077 = 40x + 27x2 Solving 1077 = 40x + 27x2 Solving for variable 'x'. Reorder the terms: 1077 + -40x + -27x2 = 40x + -40x + 27x2 + -27x2 Combine like terms: 40x + -40x = 0 1077 + -40x + -27x2 = 0 + 27x2 + -27x2 1077 + -40x + -27x2 = 27x2 + -27x2 Combine like terms: 27x2 + -27x2 = 0 1077 + -40x + -27x2 = 0 Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -39.88888889 + 1.481481481x + x2 = 0 Move the constant term to the right: Add '39.88888889' to each side of the equation. -39.88888889 + 1.481481481x + 39.88888889 + x2 = 0 + 39.88888889 Reorder the terms: -39.88888889 + 39.88888889 + 1.481481481x + x2 = 0 + 39.88888889 Combine like terms: -39.88888889 + 39.88888889 = 0.00000000 0.00000000 + 1.481481481x + x2 = 0 + 39.88888889 1.481481481x + x2 = 0 + 39.88888889 Combine like terms: 0 + 39.88888889 = 39.88888889 1.481481481x + x2 = 39.88888889 The x term is 1.481481481x. Take half its coefficient (0.7407407405). Square it (0.5486968446) and add it to both sides. Add '0.5486968446' to each side of the equation. 1.481481481x + 0.5486968446 + x2 = 39.88888889 + 0.5486968446 Reorder the terms: 0.5486968446 + 1.481481481x + x2 = 39.88888889 + 0.5486968446 Combine like terms: 39.88888889 + 0.5486968446 = 40.4375857346 0.5486968446 + 1.481481481x + x2 = 40.4375857346 Factor a perfect square on the left side: (x + 0.7407407405)(x + 0.7407407405) = 40.4375857346 Calculate the square root of the right side: 6.359055412 Break this problem into two subproblems by setting (x + 0.7407407405) equal to 6.359055412 and -6.359055412.Subproblem 1
x + 0.7407407405 = 6.359055412 Simplifying x + 0.7407407405 = 6.359055412 Reorder the terms: 0.7407407405 + x = 6.359055412 Solving 0.7407407405 + x = 6.359055412 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7407407405' to each side of the equation. 0.7407407405 + -0.7407407405 + x = 6.359055412 + -0.7407407405 Combine like terms: 0.7407407405 + -0.7407407405 = 0.0000000000 0.0000000000 + x = 6.359055412 + -0.7407407405 x = 6.359055412 + -0.7407407405 Combine like terms: 6.359055412 + -0.7407407405 = 5.6183146715 x = 5.6183146715 Simplifying x = 5.6183146715Subproblem 2
x + 0.7407407405 = -6.359055412 Simplifying x + 0.7407407405 = -6.359055412 Reorder the terms: 0.7407407405 + x = -6.359055412 Solving 0.7407407405 + x = -6.359055412 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7407407405' to each side of the equation. 0.7407407405 + -0.7407407405 + x = -6.359055412 + -0.7407407405 Combine like terms: 0.7407407405 + -0.7407407405 = 0.0000000000 0.0000000000 + x = -6.359055412 + -0.7407407405 x = -6.359055412 + -0.7407407405 Combine like terms: -6.359055412 + -0.7407407405 = -7.0997961525 x = -7.0997961525 Simplifying x = -7.0997961525Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.6183146715, -7.0997961525}
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