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Simplifying 64 = -6t2 + 96t Reorder the terms: 64 = 96t + -6t2 Solving 64 = 96t + -6t2 Solving for variable 't'. Reorder the terms: 64 + -96t + 6t2 = 96t + -96t + -6t2 + 6t2 Combine like terms: 96t + -96t = 0 64 + -96t + 6t2 = 0 + -6t2 + 6t2 64 + -96t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 64 + -96t + 6t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(32 + -48t + 3t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(32 + -48t + 3t2)' equal to zero and attempt to solve: Simplifying 32 + -48t + 3t2 = 0 Solving 32 + -48t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 10.66666667 + -16t + t2 = 0 Move the constant term to the right: Add '-10.66666667' to each side of the equation. 10.66666667 + -16t + -10.66666667 + t2 = 0 + -10.66666667 Reorder the terms: 10.66666667 + -10.66666667 + -16t + t2 = 0 + -10.66666667 Combine like terms: 10.66666667 + -10.66666667 = 0.00000000 0.00000000 + -16t + t2 = 0 + -10.66666667 -16t + t2 = 0 + -10.66666667 Combine like terms: 0 + -10.66666667 = -10.66666667 -16t + t2 = -10.66666667 The t term is -16t. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16t + 64 + t2 = -10.66666667 + 64 Reorder the terms: 64 + -16t + t2 = -10.66666667 + 64 Combine like terms: -10.66666667 + 64 = 53.33333333 64 + -16t + t2 = 53.33333333 Factor a perfect square on the left side: (t + -8)(t + -8) = 53.33333333 Calculate the square root of the right side: 7.302967433 Break this problem into two subproblems by setting (t + -8) equal to 7.302967433 and -7.302967433.Subproblem 1
t + -8 = 7.302967433 Simplifying t + -8 = 7.302967433 Reorder the terms: -8 + t = 7.302967433 Solving -8 + t = 7.302967433 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = 7.302967433 + 8 Combine like terms: -8 + 8 = 0 0 + t = 7.302967433 + 8 t = 7.302967433 + 8 Combine like terms: 7.302967433 + 8 = 15.302967433 t = 15.302967433 Simplifying t = 15.302967433Subproblem 2
t + -8 = -7.302967433 Simplifying t + -8 = -7.302967433 Reorder the terms: -8 + t = -7.302967433 Solving -8 + t = -7.302967433 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + t = -7.302967433 + 8 Combine like terms: -8 + 8 = 0 0 + t = -7.302967433 + 8 t = -7.302967433 + 8 Combine like terms: -7.302967433 + 8 = 0.697032567 t = 0.697032567 Simplifying t = 0.697032567Solution
The solution to the problem is based on the solutions from the subproblems. t = {15.302967433, 0.697032567}Solution
t = {15.302967433, 0.697032567}
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