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Simplifying 34(t) = -16t2 + 80t + 5 Reorder the terms: 34t = 5 + 80t + -16t2 Solving 34t = 5 + 80t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 34t + -80t + 16t2 = 5 + 80t + -16t2 + -5 + -80t + 16t2 Combine like terms: 34t + -80t = -46t -5 + -46t + 16t2 = 5 + 80t + -16t2 + -5 + -80t + 16t2 Reorder the terms: -5 + -46t + 16t2 = 5 + -5 + 80t + -80t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + -46t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 -5 + -46t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 -5 + -46t + 16t2 = 0 + -16t2 + 16t2 -5 + -46t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + -46t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + -2.875t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + -2.875t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + -2.875t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + -2.875t + t2 = 0 + 0.3125 -2.875t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 -2.875t + t2 = 0.3125 The t term is -2.875t. Take half its coefficient (-1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. -2.875t + 2.06640625 + t2 = 0.3125 + 2.06640625 Reorder the terms: 2.06640625 + -2.875t + t2 = 0.3125 + 2.06640625 Combine like terms: 0.3125 + 2.06640625 = 2.37890625 2.06640625 + -2.875t + t2 = 2.37890625 Factor a perfect square on the left side: (t + -1.4375)(t + -1.4375) = 2.37890625 Calculate the square root of the right side: 1.542370335 Break this problem into two subproblems by setting (t + -1.4375) equal to 1.542370335 and -1.542370335.Subproblem 1
t + -1.4375 = 1.542370335 Simplifying t + -1.4375 = 1.542370335 Reorder the terms: -1.4375 + t = 1.542370335 Solving -1.4375 + t = 1.542370335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = 1.542370335 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = 1.542370335 + 1.4375 t = 1.542370335 + 1.4375 Combine like terms: 1.542370335 + 1.4375 = 2.979870335 t = 2.979870335 Simplifying t = 2.979870335Subproblem 2
t + -1.4375 = -1.542370335 Simplifying t + -1.4375 = -1.542370335 Reorder the terms: -1.4375 + t = -1.542370335 Solving -1.4375 + t = -1.542370335 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.4375' to each side of the equation. -1.4375 + 1.4375 + t = -1.542370335 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + t = -1.542370335 + 1.4375 t = -1.542370335 + 1.4375 Combine like terms: -1.542370335 + 1.4375 = -0.104870335 t = -0.104870335 Simplifying t = -0.104870335Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.979870335, -0.104870335}
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