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Simplifying 0 = 4.905t2 + -19t + 11 Reorder the terms: 0 = 11 + -19t + 4.905t2 Solving 0 = 11 + -19t + 4.905t2 Solving for variable 't'. Combine like terms: 0 + -11 = -11 -11 + 19t + -4.905t2 = 11 + -19t + 4.905t2 + -11 + 19t + -4.905t2 Reorder the terms: -11 + 19t + -4.905t2 = 11 + -11 + -19t + 19t + 4.905t2 + -4.905t2 Combine like terms: 11 + -11 = 0 -11 + 19t + -4.905t2 = 0 + -19t + 19t + 4.905t2 + -4.905t2 -11 + 19t + -4.905t2 = -19t + 19t + 4.905t2 + -4.905t2 Combine like terms: -19t + 19t = 0 -11 + 19t + -4.905t2 = 0 + 4.905t2 + -4.905t2 -11 + 19t + -4.905t2 = 4.905t2 + -4.905t2 Combine like terms: 4.905t2 + -4.905t2 = 0.000 -11 + 19t + -4.905t2 = 0.000 Begin completing the square. Divide all terms by -4.905 the coefficient of the squared term: Divide each side by '-4.905'. 2.242609582 + -3.873598369t + t2 = 0 Move the constant term to the right: Add '-2.242609582' to each side of the equation. 2.242609582 + -3.873598369t + -2.242609582 + t2 = 0 + -2.242609582 Reorder the terms: 2.242609582 + -2.242609582 + -3.873598369t + t2 = 0 + -2.242609582 Combine like terms: 2.242609582 + -2.242609582 = 0.000000000 0.000000000 + -3.873598369t + t2 = 0 + -2.242609582 -3.873598369t + t2 = 0 + -2.242609582 Combine like terms: 0 + -2.242609582 = -2.242609582 -3.873598369t + t2 = -2.242609582 The t term is -3.873598369t. Take half its coefficient (-1.936799185). Square it (3.751191083) and add it to both sides. Add '3.751191083' to each side of the equation. -3.873598369t + 3.751191083 + t2 = -2.242609582 + 3.751191083 Reorder the terms: 3.751191083 + -3.873598369t + t2 = -2.242609582 + 3.751191083 Combine like terms: -2.242609582 + 3.751191083 = 1.508581501 3.751191083 + -3.873598369t + t2 = 1.508581501 Factor a perfect square on the left side: (t + -1.936799185)(t + -1.936799185) = 1.508581501 Calculate the square root of the right side: 1.228243258 Break this problem into two subproblems by setting (t + -1.936799185) equal to 1.228243258 and -1.228243258.Subproblem 1
t + -1.936799185 = 1.228243258 Simplifying t + -1.936799185 = 1.228243258 Reorder the terms: -1.936799185 + t = 1.228243258 Solving -1.936799185 + t = 1.228243258 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.936799185' to each side of the equation. -1.936799185 + 1.936799185 + t = 1.228243258 + 1.936799185 Combine like terms: -1.936799185 + 1.936799185 = 0.000000000 0.000000000 + t = 1.228243258 + 1.936799185 t = 1.228243258 + 1.936799185 Combine like terms: 1.228243258 + 1.936799185 = 3.165042443 t = 3.165042443 Simplifying t = 3.165042443Subproblem 2
t + -1.936799185 = -1.228243258 Simplifying t + -1.936799185 = -1.228243258 Reorder the terms: -1.936799185 + t = -1.228243258 Solving -1.936799185 + t = -1.228243258 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.936799185' to each side of the equation. -1.936799185 + 1.936799185 + t = -1.228243258 + 1.936799185 Combine like terms: -1.936799185 + 1.936799185 = 0.000000000 0.000000000 + t = -1.228243258 + 1.936799185 t = -1.228243258 + 1.936799185 Combine like terms: -1.228243258 + 1.936799185 = 0.708555927 t = 0.708555927 Simplifying t = 0.708555927Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.165042443, 0.708555927}
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