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Simplifying 16t2 + -120t = -180 Reorder the terms: -120t + 16t2 = -180 Solving -120t + 16t2 = -180 Solving for variable 't'. Reorder the terms: 180 + -120t + 16t2 = -180 + 180 Combine like terms: -180 + 180 = 0 180 + -120t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(45 + -30t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(45 + -30t + 4t2)' equal to zero and attempt to solve: Simplifying 45 + -30t + 4t2 = 0 Solving 45 + -30t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 11.25 + -7.5t + t2 = 0 Move the constant term to the right: Add '-11.25' to each side of the equation. 11.25 + -7.5t + -11.25 + t2 = 0 + -11.25 Reorder the terms: 11.25 + -11.25 + -7.5t + t2 = 0 + -11.25 Combine like terms: 11.25 + -11.25 = 0.00 0.00 + -7.5t + t2 = 0 + -11.25 -7.5t + t2 = 0 + -11.25 Combine like terms: 0 + -11.25 = -11.25 -7.5t + t2 = -11.25 The t term is -7.5t. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5t + 14.0625 + t2 = -11.25 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -11.25 + 14.0625 Combine like terms: -11.25 + 14.0625 = 2.8125 14.0625 + -7.5t + t2 = 2.8125 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (t + -3.75) equal to 1.677050983 and -1.677050983.Subproblem 1
t + -3.75 = 1.677050983 Simplifying t + -3.75 = 1.677050983 Reorder the terms: -3.75 + t = 1.677050983 Solving -3.75 + t = 1.677050983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = 1.677050983 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 1.677050983 + 3.75 t = 1.677050983 + 3.75 Combine like terms: 1.677050983 + 3.75 = 5.427050983 t = 5.427050983 Simplifying t = 5.427050983Subproblem 2
t + -3.75 = -1.677050983 Simplifying t + -3.75 = -1.677050983 Reorder the terms: -3.75 + t = -1.677050983 Solving -3.75 + t = -1.677050983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + t = -1.677050983 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -1.677050983 + 3.75 t = -1.677050983 + 3.75 Combine like terms: -1.677050983 + 3.75 = 2.072949017 t = 2.072949017 Simplifying t = 2.072949017Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.427050983, 2.072949017}Solution
t = {5.427050983, 2.072949017}
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