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Simplifying 10 = -1t2 + 20t + -20 Reorder the terms: 10 = -20 + 20t + -1t2 Solving 10 = -20 + 20t + -1t2 Solving for variable 't'. Combine like terms: 10 + 20 = 30 30 + -20t + t2 = -20 + 20t + -1t2 + 20 + -20t + t2 Reorder the terms: 30 + -20t + t2 = -20 + 20 + 20t + -20t + -1t2 + t2 Combine like terms: -20 + 20 = 0 30 + -20t + t2 = 0 + 20t + -20t + -1t2 + t2 30 + -20t + t2 = 20t + -20t + -1t2 + t2 Combine like terms: 20t + -20t = 0 30 + -20t + t2 = 0 + -1t2 + t2 30 + -20t + t2 = -1t2 + t2 Combine like terms: -1t2 + t2 = 0 30 + -20t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-30' to each side of the equation. 30 + -20t + -30 + t2 = 0 + -30 Reorder the terms: 30 + -30 + -20t + t2 = 0 + -30 Combine like terms: 30 + -30 = 0 0 + -20t + t2 = 0 + -30 -20t + t2 = 0 + -30 Combine like terms: 0 + -30 = -30 -20t + t2 = -30 The t term is -20t. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20t + 100 + t2 = -30 + 100 Reorder the terms: 100 + -20t + t2 = -30 + 100 Combine like terms: -30 + 100 = 70 100 + -20t + t2 = 70 Factor a perfect square on the left side: (t + -10)(t + -10) = 70 Calculate the square root of the right side: 8.366600265 Break this problem into two subproblems by setting (t + -10) equal to 8.366600265 and -8.366600265.Subproblem 1
t + -10 = 8.366600265 Simplifying t + -10 = 8.366600265 Reorder the terms: -10 + t = 8.366600265 Solving -10 + t = 8.366600265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + t = 8.366600265 + 10 Combine like terms: -10 + 10 = 0 0 + t = 8.366600265 + 10 t = 8.366600265 + 10 Combine like terms: 8.366600265 + 10 = 18.366600265 t = 18.366600265 Simplifying t = 18.366600265Subproblem 2
t + -10 = -8.366600265 Simplifying t + -10 = -8.366600265 Reorder the terms: -10 + t = -8.366600265 Solving -10 + t = -8.366600265 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + t = -8.366600265 + 10 Combine like terms: -10 + 10 = 0 0 + t = -8.366600265 + 10 t = -8.366600265 + 10 Combine like terms: -8.366600265 + 10 = 1.633399735 t = 1.633399735 Simplifying t = 1.633399735Solution
The solution to the problem is based on the solutions from the subproblems. t = {18.366600265, 1.633399735}
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