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Simplifying 18x2 + 27x + 5 = 0 Reorder the terms: 5 + 27x + 18x2 = 0 Solving 5 + 27x + 18x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.2777777778 + 1.5x + x2 = 0 Move the constant term to the right: Add '-0.2777777778' to each side of the equation. 0.2777777778 + 1.5x + -0.2777777778 + x2 = 0 + -0.2777777778 Reorder the terms: 0.2777777778 + -0.2777777778 + 1.5x + x2 = 0 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + 1.5x + x2 = 0 + -0.2777777778 1.5x + x2 = 0 + -0.2777777778 Combine like terms: 0 + -0.2777777778 = -0.2777777778 1.5x + x2 = -0.2777777778 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = -0.2777777778 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = -0.2777777778 + 0.5625 Combine like terms: -0.2777777778 + 0.5625 = 0.2847222222 0.5625 + 1.5x + x2 = 0.2847222222 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 0.2847222222 Calculate the square root of the right side: 0.533593686 Break this problem into two subproblems by setting (x + 0.75) equal to 0.533593686 and -0.533593686.Subproblem 1
x + 0.75 = 0.533593686 Simplifying x + 0.75 = 0.533593686 Reorder the terms: 0.75 + x = 0.533593686 Solving 0.75 + x = 0.533593686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 0.533593686 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.533593686 + -0.75 x = 0.533593686 + -0.75 Combine like terms: 0.533593686 + -0.75 = -0.216406314 x = -0.216406314 Simplifying x = -0.216406314Subproblem 2
x + 0.75 = -0.533593686 Simplifying x + 0.75 = -0.533593686 Reorder the terms: 0.75 + x = -0.533593686 Solving 0.75 + x = -0.533593686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -0.533593686 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.533593686 + -0.75 x = -0.533593686 + -0.75 Combine like terms: -0.533593686 + -0.75 = -1.283593686 x = -1.283593686 Simplifying x = -1.283593686Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.216406314, -1.283593686}
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