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Simplifying 5(t) = -16t2 + 40t + 5 Reorder the terms: 5t = 5 + 40t + -16t2 Solving 5t = 5 + 40t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 5t + -40t + 16t2 = 5 + 40t + -16t2 + -5 + -40t + 16t2 Combine like terms: 5t + -40t = -35t -5 + -35t + 16t2 = 5 + 40t + -16t2 + -5 + -40t + 16t2 Reorder the terms: -5 + -35t + 16t2 = 5 + -5 + 40t + -40t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -35t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 -5 + -35t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -5 + -35t + 16t2 = 0 + -16t2 + 16t2 -5 + -35t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -35t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -2.1875t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -2.1875t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -2.1875t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -2.1875t + t2 = 0 + 0.3125 -2.1875t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -2.1875t + t2 = 0.3125 The t term is -2.1875t. Take half its coefficient (-1.09375). Square it (1.196289063) and add it to both sides. Add '1.196289063' to each side of the equation. -2.1875t + 1.196289063 + t2 = 0.3125 + 1.196289063 Reorder the terms: 1.196289063 + -2.1875t + t2 = 0.3125 + 1.196289063 Combine like terms: 0.3125 + 1.196289063 = 1.508789063 1.196289063 + -2.1875t + t2 = 1.508789063 Factor a perfect square on the left side: (t + -1.09375)(t + -1.09375) = 1.508789063 Calculate the square root of the right side: 1.228327751 Break this problem into two subproblems by setting (t + -1.09375) equal to 1.228327751 and -1.228327751.Subproblem 1
t + -1.09375 = 1.228327751 Simplifying t + -1.09375 = 1.228327751 Reorder the terms: -1.09375 + t = 1.228327751 Solving -1.09375 + t = 1.228327751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = 1.228327751 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = 1.228327751 + 1.09375 t = 1.228327751 + 1.09375 Combine like terms: 1.228327751 + 1.09375 = 2.322077751 t = 2.322077751 Simplifying t = 2.322077751Subproblem 2
t + -1.09375 = -1.228327751 Simplifying t + -1.09375 = -1.228327751 Reorder the terms: -1.09375 + t = -1.228327751 Solving -1.09375 + t = -1.228327751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + t = -1.228327751 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + t = -1.228327751 + 1.09375 t = -1.228327751 + 1.09375 Combine like terms: -1.228327751 + 1.09375 = -0.134577751 t = -0.134577751 Simplifying t = -0.134577751Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.322077751, -0.134577751}
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