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Simplifying 84 = -16t2 + 76t + 3 Reorder the terms: 84 = 3 + 76t + -16t2 Solving 84 = 3 + 76t + -16t2 Solving for variable 't'. Combine like terms: 84 + -3 = 81 81 + -76t + 16t2 = 3 + 76t + -16t2 + -3 + -76t + 16t2 Reorder the terms: 81 + -76t + 16t2 = 3 + -3 + 76t + -76t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 81 + -76t + 16t2 = 0 + 76t + -76t + -16t2 + 16t2 81 + -76t + 16t2 = 76t + -76t + -16t2 + 16t2 Combine like terms: 76t + -76t = 0 81 + -76t + 16t2 = 0 + -16t2 + 16t2 81 + -76t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 81 + -76t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 5.0625 + -4.75t + t2 = 0 Move the constant term to the right: Add '-5.0625' to each side of the equation. 5.0625 + -4.75t + -5.0625 + t2 = 0 + -5.0625 Reorder the terms: 5.0625 + -5.0625 + -4.75t + t2 = 0 + -5.0625 Combine like terms: 5.0625 + -5.0625 = 0.0000 0.0000 + -4.75t + t2 = 0 + -5.0625 -4.75t + t2 = 0 + -5.0625 Combine like terms: 0 + -5.0625 = -5.0625 -4.75t + t2 = -5.0625 The t term is -4.75t. Take half its coefficient (-2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. -4.75t + 5.640625 + t2 = -5.0625 + 5.640625 Reorder the terms: 5.640625 + -4.75t + t2 = -5.0625 + 5.640625 Combine like terms: -5.0625 + 5.640625 = 0.578125 5.640625 + -4.75t + t2 = 0.578125 Factor a perfect square on the left side: (t + -2.375)(t + -2.375) = 0.578125 Calculate the square root of the right side: 0.760345316 Break this problem into two subproblems by setting (t + -2.375) equal to 0.760345316 and -0.760345316.Subproblem 1
t + -2.375 = 0.760345316 Simplifying t + -2.375 = 0.760345316 Reorder the terms: -2.375 + t = 0.760345316 Solving -2.375 + t = 0.760345316 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = 0.760345316 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = 0.760345316 + 2.375 t = 0.760345316 + 2.375 Combine like terms: 0.760345316 + 2.375 = 3.135345316 t = 3.135345316 Simplifying t = 3.135345316Subproblem 2
t + -2.375 = -0.760345316 Simplifying t + -2.375 = -0.760345316 Reorder the terms: -2.375 + t = -0.760345316 Solving -2.375 + t = -0.760345316 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.375' to each side of the equation. -2.375 + 2.375 + t = -0.760345316 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + t = -0.760345316 + 2.375 t = -0.760345316 + 2.375 Combine like terms: -0.760345316 + 2.375 = 1.614654684 t = 1.614654684 Simplifying t = 1.614654684Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.135345316, 1.614654684}
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