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Simplifying 0 = 8.14 + 10t + -1.86t2 Solving 0 = 8.14 + 10t + -1.86t2 Solving for variable 't'. Combine like terms: 0 + -8.14 = -8.14 -8.14 + -10t + 1.86t2 = 8.14 + 10t + -1.86t2 + -8.14 + -10t + 1.86t2 Reorder the terms: -8.14 + -10t + 1.86t2 = 8.14 + -8.14 + 10t + -10t + -1.86t2 + 1.86t2 Combine like terms: 8.14 + -8.14 = 0.00 -8.14 + -10t + 1.86t2 = 0.00 + 10t + -10t + -1.86t2 + 1.86t2 -8.14 + -10t + 1.86t2 = 10t + -10t + -1.86t2 + 1.86t2 Combine like terms: 10t + -10t = 0 -8.14 + -10t + 1.86t2 = 0 + -1.86t2 + 1.86t2 -8.14 + -10t + 1.86t2 = -1.86t2 + 1.86t2 Combine like terms: -1.86t2 + 1.86t2 = 0.00 -8.14 + -10t + 1.86t2 = 0.00 Begin completing the square. Divide all terms by 1.86 the coefficient of the squared term: Divide each side by '1.86'. -4.376344086 + -5.376344086t + t2 = 0 Move the constant term to the right: Add '4.376344086' to each side of the equation. -4.376344086 + -5.376344086t + 4.376344086 + t2 = 0 + 4.376344086 Reorder the terms: -4.376344086 + 4.376344086 + -5.376344086t + t2 = 0 + 4.376344086 Combine like terms: -4.376344086 + 4.376344086 = 0.000000000 0.000000000 + -5.376344086t + t2 = 0 + 4.376344086 -5.376344086t + t2 = 0 + 4.376344086 Combine like terms: 0 + 4.376344086 = 4.376344086 -5.376344086t + t2 = 4.376344086 The t term is -5.376344086t. Take half its coefficient (-2.688172043). Square it (7.226268933) and add it to both sides. Add '7.226268933' to each side of the equation. -5.376344086t + 7.226268933 + t2 = 4.376344086 + 7.226268933 Reorder the terms: 7.226268933 + -5.376344086t + t2 = 4.376344086 + 7.226268933 Combine like terms: 4.376344086 + 7.226268933 = 11.602613019 7.226268933 + -5.376344086t + t2 = 11.602613019 Factor a perfect square on the left side: (t + -2.688172043)(t + -2.688172043) = 11.602613019 Calculate the square root of the right side: 3.406260856 Break this problem into two subproblems by setting (t + -2.688172043) equal to 3.406260856 and -3.406260856.Subproblem 1
t + -2.688172043 = 3.406260856 Simplifying t + -2.688172043 = 3.406260856 Reorder the terms: -2.688172043 + t = 3.406260856 Solving -2.688172043 + t = 3.406260856 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.688172043' to each side of the equation. -2.688172043 + 2.688172043 + t = 3.406260856 + 2.688172043 Combine like terms: -2.688172043 + 2.688172043 = 0.000000000 0.000000000 + t = 3.406260856 + 2.688172043 t = 3.406260856 + 2.688172043 Combine like terms: 3.406260856 + 2.688172043 = 6.094432899 t = 6.094432899 Simplifying t = 6.094432899Subproblem 2
t + -2.688172043 = -3.406260856 Simplifying t + -2.688172043 = -3.406260856 Reorder the terms: -2.688172043 + t = -3.406260856 Solving -2.688172043 + t = -3.406260856 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.688172043' to each side of the equation. -2.688172043 + 2.688172043 + t = -3.406260856 + 2.688172043 Combine like terms: -2.688172043 + 2.688172043 = 0.000000000 0.000000000 + t = -3.406260856 + 2.688172043 t = -3.406260856 + 2.688172043 Combine like terms: -3.406260856 + 2.688172043 = -0.718088813 t = -0.718088813 Simplifying t = -0.718088813Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.094432899, -0.718088813}
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