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Simplifying 10000 = 50 + 77t + 3t2 Solving 10000 = 50 + 77t + 3t2 Solving for variable 't'. Combine like terms: 10000 + -50 = 9950 9950 + -77t + -3t2 = 50 + 77t + 3t2 + -50 + -77t + -3t2 Reorder the terms: 9950 + -77t + -3t2 = 50 + -50 + 77t + -77t + 3t2 + -3t2 Combine like terms: 50 + -50 = 0 9950 + -77t + -3t2 = 0 + 77t + -77t + 3t2 + -3t2 9950 + -77t + -3t2 = 77t + -77t + 3t2 + -3t2 Combine like terms: 77t + -77t = 0 9950 + -77t + -3t2 = 0 + 3t2 + -3t2 9950 + -77t + -3t2 = 3t2 + -3t2 Combine like terms: 3t2 + -3t2 = 0 9950 + -77t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -3316.666667 + 25.66666667t + t2 = 0 Move the constant term to the right: Add '3316.666667' to each side of the equation. -3316.666667 + 25.66666667t + 3316.666667 + t2 = 0 + 3316.666667 Reorder the terms: -3316.666667 + 3316.666667 + 25.66666667t + t2 = 0 + 3316.666667 Combine like terms: -3316.666667 + 3316.666667 = 0.000000 0.000000 + 25.66666667t + t2 = 0 + 3316.666667 25.66666667t + t2 = 0 + 3316.666667 Combine like terms: 0 + 3316.666667 = 3316.666667 25.66666667t + t2 = 3316.666667 The t term is 25.66666667t. Take half its coefficient (12.83333334). Square it (164.6944446) and add it to both sides. Add '164.6944446' to each side of the equation. 25.66666667t + 164.6944446 + t2 = 3316.666667 + 164.6944446 Reorder the terms: 164.6944446 + 25.66666667t + t2 = 3316.666667 + 164.6944446 Combine like terms: 3316.666667 + 164.6944446 = 3481.3611116 164.6944446 + 25.66666667t + t2 = 3481.3611116 Factor a perfect square on the left side: (t + 12.83333334)(t + 12.83333334) = 3481.3611116 Calculate the square root of the right side: 59.003060188 Break this problem into two subproblems by setting (t + 12.83333334) equal to 59.003060188 and -59.003060188.Subproblem 1
t + 12.83333334 = 59.003060188 Simplifying t + 12.83333334 = 59.003060188 Reorder the terms: 12.83333334 + t = 59.003060188 Solving 12.83333334 + t = 59.003060188 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + t = 59.003060188 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + t = 59.003060188 + -12.83333334 t = 59.003060188 + -12.83333334 Combine like terms: 59.003060188 + -12.83333334 = 46.169726848 t = 46.169726848 Simplifying t = 46.169726848Subproblem 2
t + 12.83333334 = -59.003060188 Simplifying t + 12.83333334 = -59.003060188 Reorder the terms: 12.83333334 + t = -59.003060188 Solving 12.83333334 + t = -59.003060188 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + t = -59.003060188 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + t = -59.003060188 + -12.83333334 t = -59.003060188 + -12.83333334 Combine like terms: -59.003060188 + -12.83333334 = -71.836393528 t = -71.836393528 Simplifying t = -71.836393528Solution
The solution to the problem is based on the solutions from the subproblems. t = {46.169726848, -71.836393528}
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