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Simplifying 0.07x + 0.09x(x + -1500) = 409 Reorder the terms: 0.07x + 0.09x(-1500 + x) = 409 0.07x + (-1500 * 0.09x + x * 0.09x) = 409 0.07x + (-135x + 0.09x2) = 409 Combine like terms: 0.07x + -135x = -134.93x -134.93x + 0.09x2 = 409 Solving -134.93x + 0.09x2 = 409 Solving for variable 'x'. Reorder the terms: -409 + -134.93x + 0.09x2 = 409 + -409 Combine like terms: 409 + -409 = 0 -409 + -134.93x + 0.09x2 = 0 Begin completing the square. Divide all terms by 0.09 the coefficient of the squared term: Divide each side by '0.09'. -4544.444444 + -1499.222222x + x2 = 0 Move the constant term to the right: Add '4544.444444' to each side of the equation. -4544.444444 + -1499.222222x + 4544.444444 + x2 = 0 + 4544.444444 Reorder the terms: -4544.444444 + 4544.444444 + -1499.222222x + x2 = 0 + 4544.444444 Combine like terms: -4544.444444 + 4544.444444 = 0.000000 0.000000 + -1499.222222x + x2 = 0 + 4544.444444 -1499.222222x + x2 = 0 + 4544.444444 Combine like terms: 0 + 4544.444444 = 4544.444444 -1499.222222x + x2 = 4544.444444 The x term is -1499.222222x. Take half its coefficient (-749.611111). Square it (561916.8177) and add it to both sides. Add '561916.8177' to each side of the equation. -1499.222222x + 561916.8177 + x2 = 4544.444444 + 561916.8177 Reorder the terms: 561916.8177 + -1499.222222x + x2 = 4544.444444 + 561916.8177 Combine like terms: 4544.444444 + 561916.8177 = 566461.262144 561916.8177 + -1499.222222x + x2 = 566461.262144 Factor a perfect square on the left side: (x + -749.611111)(x + -749.611111) = 566461.262144 Calculate the square root of the right side: 752.636208366 Break this problem into two subproblems by setting (x + -749.611111) equal to 752.636208366 and -752.636208366.Subproblem 1
x + -749.611111 = 752.636208366 Simplifying x + -749.611111 = 752.636208366 Reorder the terms: -749.611111 + x = 752.636208366 Solving -749.611111 + x = 752.636208366 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '749.611111' to each side of the equation. -749.611111 + 749.611111 + x = 752.636208366 + 749.611111 Combine like terms: -749.611111 + 749.611111 = 0.000000 0.000000 + x = 752.636208366 + 749.611111 x = 752.636208366 + 749.611111 Combine like terms: 752.636208366 + 749.611111 = 1502.247319366 x = 1502.247319366 Simplifying x = 1502.247319366Subproblem 2
x + -749.611111 = -752.636208366 Simplifying x + -749.611111 = -752.636208366 Reorder the terms: -749.611111 + x = -752.636208366 Solving -749.611111 + x = -752.636208366 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '749.611111' to each side of the equation. -749.611111 + 749.611111 + x = -752.636208366 + 749.611111 Combine like terms: -749.611111 + 749.611111 = 0.000000 0.000000 + x = -752.636208366 + 749.611111 x = -752.636208366 + 749.611111 Combine like terms: -752.636208366 + 749.611111 = -3.025097366 x = -3.025097366 Simplifying x = -3.025097366Solution
The solution to the problem is based on the solutions from the subproblems. x = {1502.247319366, -3.025097366}
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