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Simplifying 0 = -16t2 + 67t + 3 Reorder the terms: 0 = 3 + 67t + -16t2 Solving 0 = 3 + 67t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + -67t + 16t2 = 3 + 67t + -16t2 + -3 + -67t + 16t2 Reorder the terms: -3 + -67t + 16t2 = 3 + -3 + 67t + -67t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 -3 + -67t + 16t2 = 0 + 67t + -67t + -16t2 + 16t2 -3 + -67t + 16t2 = 67t + -67t + -16t2 + 16t2 Combine like terms: 67t + -67t = 0 -3 + -67t + 16t2 = 0 + -16t2 + 16t2 -3 + -67t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -67t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -4.1875t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -4.1875t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -4.1875t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -4.1875t + t2 = 0 + 0.1875 -4.1875t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -4.1875t + t2 = 0.1875 The t term is -4.1875t. Take half its coefficient (-2.09375). Square it (4.383789063) and add it to both sides. Add '4.383789063' to each side of the equation. -4.1875t + 4.383789063 + t2 = 0.1875 + 4.383789063 Reorder the terms: 4.383789063 + -4.1875t + t2 = 0.1875 + 4.383789063 Combine like terms: 0.1875 + 4.383789063 = 4.571289063 4.383789063 + -4.1875t + t2 = 4.571289063 Factor a perfect square on the left side: (t + -2.09375)(t + -2.09375) = 4.571289063 Calculate the square root of the right side: 2.138057311 Break this problem into two subproblems by setting (t + -2.09375) equal to 2.138057311 and -2.138057311.Subproblem 1
t + -2.09375 = 2.138057311 Simplifying t + -2.09375 = 2.138057311 Reorder the terms: -2.09375 + t = 2.138057311 Solving -2.09375 + t = 2.138057311 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.09375' to each side of the equation. -2.09375 + 2.09375 + t = 2.138057311 + 2.09375 Combine like terms: -2.09375 + 2.09375 = 0.00000 0.00000 + t = 2.138057311 + 2.09375 t = 2.138057311 + 2.09375 Combine like terms: 2.138057311 + 2.09375 = 4.231807311 t = 4.231807311 Simplifying t = 4.231807311Subproblem 2
t + -2.09375 = -2.138057311 Simplifying t + -2.09375 = -2.138057311 Reorder the terms: -2.09375 + t = -2.138057311 Solving -2.09375 + t = -2.138057311 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.09375' to each side of the equation. -2.09375 + 2.09375 + t = -2.138057311 + 2.09375 Combine like terms: -2.09375 + 2.09375 = 0.00000 0.00000 + t = -2.138057311 + 2.09375 t = -2.138057311 + 2.09375 Combine like terms: -2.138057311 + 2.09375 = -0.044307311 t = -0.044307311 Simplifying t = -0.044307311Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.231807311, -0.044307311}
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