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Simplifying 0 = -16t2 + 192t + 30 Reorder the terms: 0 = 30 + 192t + -16t2 Solving 0 = 30 + 192t + -16t2 Solving for variable 't'. Combine like terms: 0 + -30 = -30 -30 + -192t + 16t2 = 30 + 192t + -16t2 + -30 + -192t + 16t2 Reorder the terms: -30 + -192t + 16t2 = 30 + -30 + 192t + -192t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 -30 + -192t + 16t2 = 0 + 192t + -192t + -16t2 + 16t2 -30 + -192t + 16t2 = 192t + -192t + -16t2 + 16t2 Combine like terms: 192t + -192t = 0 -30 + -192t + 16t2 = 0 + -16t2 + 16t2 -30 + -192t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -30 + -192t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + -96t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + -96t + 8t2)' equal to zero and attempt to solve: Simplifying -15 + -96t + 8t2 = 0 Solving -15 + -96t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.875 + -12t + t2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + -12t + 1.875 + t2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + -12t + t2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + -12t + t2 = 0 + 1.875 -12t + t2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 -12t + t2 = 1.875 The t term is -12t. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12t + 36 + t2 = 1.875 + 36 Reorder the terms: 36 + -12t + t2 = 1.875 + 36 Combine like terms: 1.875 + 36 = 37.875 36 + -12t + t2 = 37.875 Factor a perfect square on the left side: (t + -6)(t + -6) = 37.875 Calculate the square root of the right side: 6.154266813 Break this problem into two subproblems by setting (t + -6) equal to 6.154266813 and -6.154266813.Subproblem 1
t + -6 = 6.154266813 Simplifying t + -6 = 6.154266813 Reorder the terms: -6 + t = 6.154266813 Solving -6 + t = 6.154266813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = 6.154266813 + 6 Combine like terms: -6 + 6 = 0 0 + t = 6.154266813 + 6 t = 6.154266813 + 6 Combine like terms: 6.154266813 + 6 = 12.154266813 t = 12.154266813 Simplifying t = 12.154266813Subproblem 2
t + -6 = -6.154266813 Simplifying t + -6 = -6.154266813 Reorder the terms: -6 + t = -6.154266813 Solving -6 + t = -6.154266813 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + t = -6.154266813 + 6 Combine like terms: -6 + 6 = 0 0 + t = -6.154266813 + 6 t = -6.154266813 + 6 Combine like terms: -6.154266813 + 6 = -0.154266813 t = -0.154266813 Simplifying t = -0.154266813Solution
The solution to the problem is based on the solutions from the subproblems. t = {12.154266813, -0.154266813}Solution
t = {12.154266813, -0.154266813}
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