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Simplifying (710)(0.5 + -1x)(0.05 + -1x) = x Multiply (0.5 + -1x) * (0.05 + -1x) 710(0.5(0.05 + -1x) + -1x * (0.05 + -1x)) = x 710((0.05 * 0.5 + -1x * 0.5) + -1x * (0.05 + -1x)) = x 710((0.025 + -0.5x) + -1x * (0.05 + -1x)) = x 710(0.025 + -0.5x + (0.05 * -1x + -1x * -1x)) = x 710(0.025 + -0.5x + (-0.05x + 1x2)) = x Combine like terms: -0.5x + -0.05x = -0.55x 710(0.025 + -0.55x + 1x2) = x (0.025 * 710 + -0.55x * 710 + 1x2 * 710) = x (17.75 + -390.5x + 710x2) = x Solving 17.75 + -390.5x + 710x2 = x Solving for variable 'x'. Reorder the terms: 17.75 + -390.5x + -1x + 710x2 = x + -1x Combine like terms: -390.5x + -1x = -391.5x 17.75 + -391.5x + 710x2 = x + -1x Combine like terms: x + -1x = 0 17.75 + -391.5x + 710x2 = 0 Begin completing the square. Divide all terms by 710 the coefficient of the squared term: Divide each side by '710'. 0.025 + -0.5514084507x + x2 = 0.0 Move the constant term to the right: Add '-0.025' to each side of the equation. 0.025 + -0.5514084507x + -0.025 + x2 = 0.0 + -0.025 Reorder the terms: 0.025 + -0.025 + -0.5514084507x + x2 = 0.0 + -0.025 Combine like terms: 0.025 + -0.025 = 0.000 0.000 + -0.5514084507x + x2 = 0.0 + -0.025 -0.5514084507x + x2 = 0.0 + -0.025 Combine like terms: 0.0 + -0.025 = -0.025 -0.5514084507x + x2 = -0.025 The x term is -0.5514084507x. Take half its coefficient (-0.2757042254). Square it (0.07601281990) and add it to both sides. Add '0.07601281990' to each side of the equation. -0.5514084507x + 0.07601281990 + x2 = -0.025 + 0.07601281990 Reorder the terms: 0.07601281990 + -0.5514084507x + x2 = -0.025 + 0.07601281990 Combine like terms: -0.025 + 0.07601281990 = 0.0510128199 0.07601281990 + -0.5514084507x + x2 = 0.0510128199 Factor a perfect square on the left side: (x + -0.2757042254)(x + -0.2757042254) = 0.0510128199 Calculate the square root of the right side: 0.225860178 Break this problem into two subproblems by setting (x + -0.2757042254) equal to 0.225860178 and -0.225860178.Subproblem 1
x + -0.2757042254 = 0.225860178 Simplifying x + -0.2757042254 = 0.225860178 Reorder the terms: -0.2757042254 + x = 0.225860178 Solving -0.2757042254 + x = 0.225860178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2757042254' to each side of the equation. -0.2757042254 + 0.2757042254 + x = 0.225860178 + 0.2757042254 Combine like terms: -0.2757042254 + 0.2757042254 = 0.0000000000 0.0000000000 + x = 0.225860178 + 0.2757042254 x = 0.225860178 + 0.2757042254 Combine like terms: 0.225860178 + 0.2757042254 = 0.5015644034 x = 0.5015644034 Simplifying x = 0.5015644034Subproblem 2
x + -0.2757042254 = -0.225860178 Simplifying x + -0.2757042254 = -0.225860178 Reorder the terms: -0.2757042254 + x = -0.225860178 Solving -0.2757042254 + x = -0.225860178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2757042254' to each side of the equation. -0.2757042254 + 0.2757042254 + x = -0.225860178 + 0.2757042254 Combine like terms: -0.2757042254 + 0.2757042254 = 0.0000000000 0.0000000000 + x = -0.225860178 + 0.2757042254 x = -0.225860178 + 0.2757042254 Combine like terms: -0.225860178 + 0.2757042254 = 0.0498440474 x = 0.0498440474 Simplifying x = 0.0498440474Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5015644034, 0.0498440474}
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