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Simplifying x + x[x(0.08)] = 218040 Reorder the terms for easier multiplication: x + x[0.08x] = 218040 Remove brackets around [0.08x] x + x * 0.08x = 218040 Reorder the terms for easier multiplication: x + 0.08x * x = 218040 Multiply x * x x + 0.08x2 = 218040 Solving x + 0.08x2 = 218040 Solving for variable 'x'. Reorder the terms: -218040 + x + 0.08x2 = 218040 + -218040 Combine like terms: 218040 + -218040 = 0 -218040 + x + 0.08x2 = 0 Begin completing the square. Divide all terms by 0.08 the coefficient of the squared term: Divide each side by '0.08'. -2725500 + 12.5x + x2 = 0 Move the constant term to the right: Add '2725500' to each side of the equation. -2725500 + 12.5x + 2725500 + x2 = 0 + 2725500 Reorder the terms: -2725500 + 2725500 + 12.5x + x2 = 0 + 2725500 Combine like terms: -2725500 + 2725500 = 0 0 + 12.5x + x2 = 0 + 2725500 12.5x + x2 = 0 + 2725500 Combine like terms: 0 + 2725500 = 2725500 12.5x + x2 = 2725500 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 12.5x + 0.25 + x2 = 2725500 + 0.25 Reorder the terms: 0.25 + 12.5x + x2 = 2725500 + 0.25 Combine like terms: 2725500 + 0.25 = 2725500.25 0.25 + 12.5x + x2 = 2725500.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2725500.25 Calculate the square root of the right side: 1650.908916325 Break this problem into two subproblems by setting (x + 0.5) equal to 1650.908916325 and -1650.908916325.Subproblem 1
x + 0.5 = 1650.908916325 Simplifying x + 0.5 = 1650.908916325 Reorder the terms: 0.5 + x = 1650.908916325 Solving 0.5 + x = 1650.908916325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1650.908916325 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1650.908916325 + -0.5 x = 1650.908916325 + -0.5 Combine like terms: 1650.908916325 + -0.5 = 1650.408916325 x = 1650.408916325 Simplifying x = 1650.408916325Subproblem 2
x + 0.5 = -1650.908916325 Simplifying x + 0.5 = -1650.908916325 Reorder the terms: 0.5 + x = -1650.908916325 Solving 0.5 + x = -1650.908916325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1650.908916325 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1650.908916325 + -0.5 x = -1650.908916325 + -0.5 Combine like terms: -1650.908916325 + -0.5 = -1651.408916325 x = -1651.408916325 Simplifying x = -1651.408916325Solution
The solution to the problem is based on the solutions from the subproblems. x = {1650.408916325, -1651.408916325}
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