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Simplifying 54 = 16t2 + 60t + 4 Reorder the terms: 54 = 4 + 60t + 16t2 Solving 54 = 4 + 60t + 16t2 Solving for variable 't'. Combine like terms: 54 + -4 = 50 50 + -60t + -16t2 = 4 + 60t + 16t2 + -4 + -60t + -16t2 Reorder the terms: 50 + -60t + -16t2 = 4 + -4 + 60t + -60t + 16t2 + -16t2 Combine like terms: 4 + -4 = 0 50 + -60t + -16t2 = 0 + 60t + -60t + 16t2 + -16t2 50 + -60t + -16t2 = 60t + -60t + 16t2 + -16t2 Combine like terms: 60t + -60t = 0 50 + -60t + -16t2 = 0 + 16t2 + -16t2 50 + -60t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 50 + -60t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(25 + -30t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(25 + -30t + -8t2)' equal to zero and attempt to solve: Simplifying 25 + -30t + -8t2 = 0 Solving 25 + -30t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -3.125 + 3.75t + t2 = 0 Move the constant term to the right: Add '3.125' to each side of the equation. -3.125 + 3.75t + 3.125 + t2 = 0 + 3.125 Reorder the terms: -3.125 + 3.125 + 3.75t + t2 = 0 + 3.125 Combine like terms: -3.125 + 3.125 = 0.000 0.000 + 3.75t + t2 = 0 + 3.125 3.75t + t2 = 0 + 3.125 Combine like terms: 0 + 3.125 = 3.125 3.75t + t2 = 3.125 The t term is 3.75t. Take half its coefficient (1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. 3.75t + 3.515625 + t2 = 3.125 + 3.515625 Reorder the terms: 3.515625 + 3.75t + t2 = 3.125 + 3.515625 Combine like terms: 3.125 + 3.515625 = 6.640625 3.515625 + 3.75t + t2 = 6.640625 Factor a perfect square on the left side: (t + 1.875)(t + 1.875) = 6.640625 Calculate the square root of the right side: 2.576941016 Break this problem into two subproblems by setting (t + 1.875) equal to 2.576941016 and -2.576941016.Subproblem 1
t + 1.875 = 2.576941016 Simplifying t + 1.875 = 2.576941016 Reorder the terms: 1.875 + t = 2.576941016 Solving 1.875 + t = 2.576941016 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + t = 2.576941016 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + t = 2.576941016 + -1.875 t = 2.576941016 + -1.875 Combine like terms: 2.576941016 + -1.875 = 0.701941016 t = 0.701941016 Simplifying t = 0.701941016Subproblem 2
t + 1.875 = -2.576941016 Simplifying t + 1.875 = -2.576941016 Reorder the terms: 1.875 + t = -2.576941016 Solving 1.875 + t = -2.576941016 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.875' to each side of the equation. 1.875 + -1.875 + t = -2.576941016 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + t = -2.576941016 + -1.875 t = -2.576941016 + -1.875 Combine like terms: -2.576941016 + -1.875 = -4.451941016 t = -4.451941016 Simplifying t = -4.451941016Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.701941016, -4.451941016}Solution
t = {0.701941016, -4.451941016}
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