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Simplifying 10t = -0.8t2 + 1.6t + 12 Reorder the terms: 10t = 12 + 1.6t + -0.8t2 Solving 10t = 12 + 1.6t + -0.8t2 Solving for variable 't'. Reorder the terms: -12 + 10t + -1.6t + 0.8t2 = 12 + 1.6t + -0.8t2 + -12 + -1.6t + 0.8t2 Combine like terms: 10t + -1.6t = 8.4t -12 + 8.4t + 0.8t2 = 12 + 1.6t + -0.8t2 + -12 + -1.6t + 0.8t2 Reorder the terms: -12 + 8.4t + 0.8t2 = 12 + -12 + 1.6t + -1.6t + -0.8t2 + 0.8t2 Combine like terms: 12 + -12 = 0 -12 + 8.4t + 0.8t2 = 0 + 1.6t + -1.6t + -0.8t2 + 0.8t2 -12 + 8.4t + 0.8t2 = 1.6t + -1.6t + -0.8t2 + 0.8t2 Combine like terms: 1.6t + -1.6t = 0.0 -12 + 8.4t + 0.8t2 = 0.0 + -0.8t2 + 0.8t2 -12 + 8.4t + 0.8t2 = -0.8t2 + 0.8t2 Combine like terms: -0.8t2 + 0.8t2 = 0.0 -12 + 8.4t + 0.8t2 = 0.0 Begin completing the square. Divide all terms by 0.8 the coefficient of the squared term: Divide each side by '0.8'. -15 + 10.5t + t2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 10.5t + 15 + t2 = 0 + 15 Reorder the terms: -15 + 15 + 10.5t + t2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 10.5t + t2 = 0 + 15 10.5t + t2 = 0 + 15 Combine like terms: 0 + 15 = 15 10.5t + t2 = 15 The t term is 10.5t. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5t + 27.5625 + t2 = 15 + 27.5625 Reorder the terms: 27.5625 + 10.5t + t2 = 15 + 27.5625 Combine like terms: 15 + 27.5625 = 42.5625 27.5625 + 10.5t + t2 = 42.5625 Factor a perfect square on the left side: (t + 5.25)(t + 5.25) = 42.5625 Calculate the square root of the right side: 6.523994175 Break this problem into two subproblems by setting (t + 5.25) equal to 6.523994175 and -6.523994175.Subproblem 1
t + 5.25 = 6.523994175 Simplifying t + 5.25 = 6.523994175 Reorder the terms: 5.25 + t = 6.523994175 Solving 5.25 + t = 6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = 6.523994175 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = 6.523994175 + -5.25 t = 6.523994175 + -5.25 Combine like terms: 6.523994175 + -5.25 = 1.273994175 t = 1.273994175 Simplifying t = 1.273994175Subproblem 2
t + 5.25 = -6.523994175 Simplifying t + 5.25 = -6.523994175 Reorder the terms: 5.25 + t = -6.523994175 Solving 5.25 + t = -6.523994175 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + t = -6.523994175 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + t = -6.523994175 + -5.25 t = -6.523994175 + -5.25 Combine like terms: -6.523994175 + -5.25 = -11.773994175 t = -11.773994175 Simplifying t = -11.773994175Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.273994175, -11.773994175}
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