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Simplifying -27x2 + -53x + 527.53 = 0 Reorder the terms: 527.53 + -53x + -27x2 = 0 Solving 527.53 + -53x + -27x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -27 the coefficient of the squared term: Divide each side by '-27'. -19.53814815 + 1.962962963x + x2 = 0 Move the constant term to the right: Add '19.53814815' to each side of the equation. -19.53814815 + 1.962962963x + 19.53814815 + x2 = 0 + 19.53814815 Reorder the terms: -19.53814815 + 19.53814815 + 1.962962963x + x2 = 0 + 19.53814815 Combine like terms: -19.53814815 + 19.53814815 = 0.00000000 0.00000000 + 1.962962963x + x2 = 0 + 19.53814815 1.962962963x + x2 = 0 + 19.53814815 Combine like terms: 0 + 19.53814815 = 19.53814815 1.962962963x + x2 = 19.53814815 The x term is 1.962962963x. Take half its coefficient (0.9814814815). Square it (0.9633058985) and add it to both sides. Add '0.9633058985' to each side of the equation. 1.962962963x + 0.9633058985 + x2 = 19.53814815 + 0.9633058985 Reorder the terms: 0.9633058985 + 1.962962963x + x2 = 19.53814815 + 0.9633058985 Combine like terms: 19.53814815 + 0.9633058985 = 20.5014540485 0.9633058985 + 1.962962963x + x2 = 20.5014540485 Factor a perfect square on the left side: (x + 0.9814814815)(x + 0.9814814815) = 20.5014540485 Calculate the square root of the right side: 4.527853139 Break this problem into two subproblems by setting (x + 0.9814814815) equal to 4.527853139 and -4.527853139.Subproblem 1
x + 0.9814814815 = 4.527853139 Simplifying x + 0.9814814815 = 4.527853139 Reorder the terms: 0.9814814815 + x = 4.527853139 Solving 0.9814814815 + x = 4.527853139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9814814815' to each side of the equation. 0.9814814815 + -0.9814814815 + x = 4.527853139 + -0.9814814815 Combine like terms: 0.9814814815 + -0.9814814815 = 0.0000000000 0.0000000000 + x = 4.527853139 + -0.9814814815 x = 4.527853139 + -0.9814814815 Combine like terms: 4.527853139 + -0.9814814815 = 3.5463716575 x = 3.5463716575 Simplifying x = 3.5463716575Subproblem 2
x + 0.9814814815 = -4.527853139 Simplifying x + 0.9814814815 = -4.527853139 Reorder the terms: 0.9814814815 + x = -4.527853139 Solving 0.9814814815 + x = -4.527853139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9814814815' to each side of the equation. 0.9814814815 + -0.9814814815 + x = -4.527853139 + -0.9814814815 Combine like terms: 0.9814814815 + -0.9814814815 = 0.0000000000 0.0000000000 + x = -4.527853139 + -0.9814814815 x = -4.527853139 + -0.9814814815 Combine like terms: -4.527853139 + -0.9814814815 = -5.5093346205 x = -5.5093346205 Simplifying x = -5.5093346205Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.5463716575, -5.5093346205}
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