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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 = 27.3125 14.0625 + 7.5t + t2 = 27.3125 Factor a perfect square on the left side: (t + 3.75)(t + 3.75) = 27.3125 Calculate the square root of the right side: 5.22613624 Break this problem into two subproblems by setting (t + 3.75) equal to 5.22613624 and -5.22613624.Subproblem 1
t + 3.75 = 5.22613624 Simplifying t + 3.75 = 5.22613624 Reorder the terms: 3.75 + t = 5.22613624 Solving 3.75 + t = 5.22613624 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 = 5.22613624 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + t = 5.22613624 + -3.75 t = 5.22613624 + -3.75 Combine like terms: 5.22613624 + -3.75 = 1.47613624 t = 1.47613624 Simplifying t = 1.47613624Subproblem 2
t + 3.75 = -5.22613624 Simplifying t + 3.75 = -5.22613624 Reorder the terms: 3.75 + t = -5.22613624 Solving 3.75 + t = -5.22613624 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 = -5.22613624 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + t = -5.22613624 + -3.75 t = -5.22613624 + -3.75 Combine like terms: -5.22613624 + -3.75 = -8.97613624 t = -8.97613624 Simplifying t = -8.97613624Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.47613624, -8.97613624}Solution
t = {1.47613624, -8.97613624}
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