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Simplifying 1413.98 = -2.03x2 + 77.85x + 745 Reorder the terms: 1413.98 = 745 + 77.85x + -2.03x2 Solving 1413.98 = 745 + 77.85x + -2.03x2 Solving for variable 'x'. Combine like terms: 1413.98 + -745 = 668.98 668.98 + -77.85x + 2.03x2 = 745 + 77.85x + -2.03x2 + -745 + -77.85x + 2.03x2 Reorder the terms: 668.98 + -77.85x + 2.03x2 = 745 + -745 + 77.85x + -77.85x + -2.03x2 + 2.03x2 Combine like terms: 745 + -745 = 0 668.98 + -77.85x + 2.03x2 = 0 + 77.85x + -77.85x + -2.03x2 + 2.03x2 668.98 + -77.85x + 2.03x2 = 77.85x + -77.85x + -2.03x2 + 2.03x2 Combine like terms: 77.85x + -77.85x = 0.00 668.98 + -77.85x + 2.03x2 = 0.00 + -2.03x2 + 2.03x2 668.98 + -77.85x + 2.03x2 = -2.03x2 + 2.03x2 Combine like terms: -2.03x2 + 2.03x2 = 0.00 668.98 + -77.85x + 2.03x2 = 0.00 Begin completing the square. Divide all terms by 2.03 the coefficient of the squared term: Divide each side by '2.03'. 329.546798 + -38.34975369x + x2 = 0 Move the constant term to the right: Add '-329.546798' to each side of the equation. 329.546798 + -38.34975369x + -329.546798 + x2 = 0 + -329.546798 Reorder the terms: 329.546798 + -329.546798 + -38.34975369x + x2 = 0 + -329.546798 Combine like terms: 329.546798 + -329.546798 = 0.000000 0.000000 + -38.34975369x + x2 = 0 + -329.546798 -38.34975369x + x2 = 0 + -329.546798 Combine like terms: 0 + -329.546798 = -329.546798 -38.34975369x + x2 = -329.546798 The x term is -38.34975369x. Take half its coefficient (-19.17487685). Square it (367.6759022) and add it to both sides. Add '367.6759022' to each side of the equation. -38.34975369x + 367.6759022 + x2 = -329.546798 + 367.6759022 Reorder the terms: 367.6759022 + -38.34975369x + x2 = -329.546798 + 367.6759022 Combine like terms: -329.546798 + 367.6759022 = 38.1291042 367.6759022 + -38.34975369x + x2 = 38.1291042 Factor a perfect square on the left side: (x + -19.17487685)(x + -19.17487685) = 38.1291042 Calculate the square root of the right side: 6.174876857 Break this problem into two subproblems by setting (x + -19.17487685) equal to 6.174876857 and -6.174876857.Subproblem 1
x + -19.17487685 = 6.174876857 Simplifying x + -19.17487685 = 6.174876857 Reorder the terms: -19.17487685 + x = 6.174876857 Solving -19.17487685 + x = 6.174876857 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '19.17487685' to each side of the equation. -19.17487685 + 19.17487685 + x = 6.174876857 + 19.17487685 Combine like terms: -19.17487685 + 19.17487685 = 0.00000000 0.00000000 + x = 6.174876857 + 19.17487685 x = 6.174876857 + 19.17487685 Combine like terms: 6.174876857 + 19.17487685 = 25.349753707 x = 25.349753707 Simplifying x = 25.349753707Subproblem 2
x + -19.17487685 = -6.174876857 Simplifying x + -19.17487685 = -6.174876857 Reorder the terms: -19.17487685 + x = -6.174876857 Solving -19.17487685 + x = -6.174876857 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '19.17487685' to each side of the equation. -19.17487685 + 19.17487685 + x = -6.174876857 + 19.17487685 Combine like terms: -19.17487685 + 19.17487685 = 0.00000000 0.00000000 + x = -6.174876857 + 19.17487685 x = -6.174876857 + 19.17487685 Combine like terms: -6.174876857 + 19.17487685 = 12.999999993 x = 12.999999993 Simplifying x = 12.999999993Solution
The solution to the problem is based on the solutions from the subproblems. x = {25.349753707, 12.999999993}
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