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Simplifying 106 = -16t2 + 120t + -106 Reorder the terms: 106 = -106 + 120t + -16t2 Solving 106 = -106 + 120t + -16t2 Solving for variable 't'. Combine like terms: 106 + 106 = 212 212 + -120t + 16t2 = -106 + 120t + -16t2 + 106 + -120t + 16t2 Reorder the terms: 212 + -120t + 16t2 = -106 + 106 + 120t + -120t + -16t2 + 16t2 Combine like terms: -106 + 106 = 0 212 + -120t + 16t2 = 0 + 120t + -120t + -16t2 + 16t2 212 + -120t + 16t2 = 120t + -120t + -16t2 + 16t2 Combine like terms: 120t + -120t = 0 212 + -120t + 16t2 = 0 + -16t2 + 16t2 212 + -120t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 212 + -120t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(53 + -30t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(53 + -30t + 4t2)' equal to zero and attempt to solve: Simplifying 53 + -30t + 4t2 = 0 Solving 53 + -30t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 13.25 + -7.5t + t2 = 0 Move the constant term to the right: Add '-13.25' to each side of the equation. 13.25 + -7.5t + -13.25 + t2 = 0 + -13.25 Reorder the terms: 13.25 + -13.25 + -7.5t + t2 = 0 + -13.25 Combine like terms: 13.25 + -13.25 = 0.00 0.00 + -7.5t + t2 = 0 + -13.25 -7.5t + t2 = 0 + -13.25 Combine like terms: 0 + -13.25 = -13.25 -7.5t + t2 = -13.25 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 = -13.25 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -13.25 + 14.0625 Combine like terms: -13.25 + 14.0625 = 0.8125 14.0625 + -7.5t + t2 = 0.8125 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (t + -3.75) equal to 0.901387819 and -0.901387819.Subproblem 1
t + -3.75 = 0.901387819 Simplifying t + -3.75 = 0.901387819 Reorder the terms: -3.75 + t = 0.901387819 Solving -3.75 + t = 0.901387819 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 = 0.901387819 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 0.901387819 + 3.75 t = 0.901387819 + 3.75 Combine like terms: 0.901387819 + 3.75 = 4.651387819 t = 4.651387819 Simplifying t = 4.651387819Subproblem 2
t + -3.75 = -0.901387819 Simplifying t + -3.75 = -0.901387819 Reorder the terms: -3.75 + t = -0.901387819 Solving -3.75 + t = -0.901387819 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 = -0.901387819 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -0.901387819 + 3.75 t = -0.901387819 + 3.75 Combine like terms: -0.901387819 + 3.75 = 2.848612181 t = 2.848612181 Simplifying t = 2.848612181Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.651387819, 2.848612181}Solution
t = {4.651387819, 2.848612181}
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