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Simplifying 0.07x + 0.09x(1500 + -1x) = 409 0.07x + (1500 * 0.09x + -1x * 0.09x) = 409 0.07x + (135x + -0.09x2) = 409 Combine like terms: 0.07x + 135x = 135.07x 135.07x + -0.09x2 = 409 Solving 135.07x + -0.09x2 = 409 Solving for variable 'x'. Reorder the terms: -409 + 135.07x + -0.09x2 = 409 + -409 Combine like terms: 409 + -409 = 0 -409 + 135.07x + -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 + -1500.777778x + x2 = 0 Move the constant term to the right: Add '-4544.444444' to each side of the equation. 4544.444444 + -1500.777778x + -4544.444444 + x2 = 0 + -4544.444444 Reorder the terms: 4544.444444 + -4544.444444 + -1500.777778x + x2 = 0 + -4544.444444 Combine like terms: 4544.444444 + -4544.444444 = 0.000000 0.000000 + -1500.777778x + x2 = 0 + -4544.444444 -1500.777778x + x2 = 0 + -4544.444444 Combine like terms: 0 + -4544.444444 = -4544.444444 -1500.777778x + x2 = -4544.444444 The x term is -1500.777778x. Take half its coefficient (-750.388889). Square it (563083.4847) and add it to both sides. Add '563083.4847' to each side of the equation. -1500.777778x + 563083.4847 + x2 = -4544.444444 + 563083.4847 Reorder the terms: 563083.4847 + -1500.777778x + x2 = -4544.444444 + 563083.4847 Combine like terms: -4544.444444 + 563083.4847 = 558539.040256 563083.4847 + -1500.777778x + x2 = 558539.040256 Factor a perfect square on the left side: (x + -750.388889)(x + -750.388889) = 558539.040256 Calculate the square root of the right side: 747.354695079 Break this problem into two subproblems by setting (x + -750.388889) equal to 747.354695079 and -747.354695079.Subproblem 1
x + -750.388889 = 747.354695079 Simplifying x + -750.388889 = 747.354695079 Reorder the terms: -750.388889 + x = 747.354695079 Solving -750.388889 + x = 747.354695079 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '750.388889' to each side of the equation. -750.388889 + 750.388889 + x = 747.354695079 + 750.388889 Combine like terms: -750.388889 + 750.388889 = 0.000000 0.000000 + x = 747.354695079 + 750.388889 x = 747.354695079 + 750.388889 Combine like terms: 747.354695079 + 750.388889 = 1497.743584079 x = 1497.743584079 Simplifying x = 1497.743584079Subproblem 2
x + -750.388889 = -747.354695079 Simplifying x + -750.388889 = -747.354695079 Reorder the terms: -750.388889 + x = -747.354695079 Solving -750.388889 + x = -747.354695079 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '750.388889' to each side of the equation. -750.388889 + 750.388889 + x = -747.354695079 + 750.388889 Combine like terms: -750.388889 + 750.388889 = 0.000000 0.000000 + x = -747.354695079 + 750.388889 x = -747.354695079 + 750.388889 Combine like terms: -747.354695079 + 750.388889 = 3.034193921 x = 3.034193921 Simplifying x = 3.034193921Solution
The solution to the problem is based on the solutions from the subproblems. x = {1497.743584079, 3.034193921}
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