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Simplifying 12(t) = -16t2 + 12 Reorder the terms: 12t = 12 + -16t2 Solving 12t = 12 + -16t2 Solving for variable 't'. Reorder the terms: -12 + 12t + 16t2 = 12 + -16t2 + -12 + 16t2 Reorder the terms: -12 + 12t + 16t2 = 12 + -12 + -16t2 + 16t2 Combine like terms: 12 + -12 = 0 -12 + 12t + 16t2 = 0 + -16t2 + 16t2 -12 + 12t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -12 + 12t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + 3t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + 3t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + 3t + 4t2 = 0 Solving -3 + 3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.75t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.75t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.75t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.75t + t2 = 0 + 0.75 0.75t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.75t + t2 = 0.75 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + 0.75t + t2 = 0.890625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (t + 0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
t + 0.375 = 0.943729304 Simplifying t + 0.375 = 0.943729304 Reorder the terms: 0.375 + t = 0.943729304 Solving 0.375 + t = 0.943729304 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 0.943729304 + -0.375 t = 0.943729304 + -0.375 Combine like terms: 0.943729304 + -0.375 = 0.568729304 t = 0.568729304 Simplifying t = 0.568729304Subproblem 2
t + 0.375 = -0.943729304 Simplifying t + 0.375 = -0.943729304 Reorder the terms: 0.375 + t = -0.943729304 Solving 0.375 + t = -0.943729304 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -0.943729304 + -0.375 t = -0.943729304 + -0.375 Combine like terms: -0.943729304 + -0.375 = -1.318729304 t = -1.318729304 Simplifying t = -1.318729304Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.568729304, -1.318729304}Solution
t = {0.568729304, -1.318729304}
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