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Simplifying 123 = -16t2 + 123t + 5 Reorder the terms: 123 = 5 + 123t + -16t2 Solving 123 = 5 + 123t + -16t2 Solving for variable 't'. Combine like terms: 123 + -5 = 118 118 + -123t + 16t2 = 5 + 123t + -16t2 + -5 + -123t + 16t2 Reorder the terms: 118 + -123t + 16t2 = 5 + -5 + 123t + -123t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 118 + -123t + 16t2 = 0 + 123t + -123t + -16t2 + 16t2 118 + -123t + 16t2 = 123t + -123t + -16t2 + 16t2 Combine like terms: 123t + -123t = 0 118 + -123t + 16t2 = 0 + -16t2 + 16t2 118 + -123t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 118 + -123t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 7.375 + -7.6875t + t2 = 0 Move the constant term to the right: Add '-7.375' to each side of the equation. 7.375 + -7.6875t + -7.375 + t2 = 0 + -7.375 Reorder the terms: 7.375 + -7.375 + -7.6875t + t2 = 0 + -7.375 Combine like terms: 7.375 + -7.375 = 0.000 0.000 + -7.6875t + t2 = 0 + -7.375 -7.6875t + t2 = 0 + -7.375 Combine like terms: 0 + -7.375 = -7.375 -7.6875t + t2 = -7.375 The t term is -7.6875t. Take half its coefficient (-3.84375). Square it (14.77441406) and add it to both sides. Add '14.77441406' to each side of the equation. -7.6875t + 14.77441406 + t2 = -7.375 + 14.77441406 Reorder the terms: 14.77441406 + -7.6875t + t2 = -7.375 + 14.77441406 Combine like terms: -7.375 + 14.77441406 = 7.39941406 14.77441406 + -7.6875t + t2 = 7.39941406 Factor a perfect square on the left side: (t + -3.84375)(t + -3.84375) = 7.39941406 Calculate the square root of the right side: 2.720186402 Break this problem into two subproblems by setting (t + -3.84375) equal to 2.720186402 and -2.720186402.Subproblem 1
t + -3.84375 = 2.720186402 Simplifying t + -3.84375 = 2.720186402 Reorder the terms: -3.84375 + t = 2.720186402 Solving -3.84375 + t = 2.720186402 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.84375' to each side of the equation. -3.84375 + 3.84375 + t = 2.720186402 + 3.84375 Combine like terms: -3.84375 + 3.84375 = 0.00000 0.00000 + t = 2.720186402 + 3.84375 t = 2.720186402 + 3.84375 Combine like terms: 2.720186402 + 3.84375 = 6.563936402 t = 6.563936402 Simplifying t = 6.563936402Subproblem 2
t + -3.84375 = -2.720186402 Simplifying t + -3.84375 = -2.720186402 Reorder the terms: -3.84375 + t = -2.720186402 Solving -3.84375 + t = -2.720186402 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.84375' to each side of the equation. -3.84375 + 3.84375 + t = -2.720186402 + 3.84375 Combine like terms: -3.84375 + 3.84375 = 0.00000 0.00000 + t = -2.720186402 + 3.84375 t = -2.720186402 + 3.84375 Combine like terms: -2.720186402 + 3.84375 = 1.123563598 t = 1.123563598 Simplifying t = 1.123563598Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.563936402, 1.123563598}
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