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Simplifying -16t2 + 120t + 20 = 0 Reorder the terms: 20 + 120t + -16t2 = 0 Solving 20 + 120t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(5 + 30t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(5 + 30t + -4t2)' equal to zero and attempt to solve: Simplifying 5 + 30t + -4t2 = 0 Solving 5 + 30t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.25 + -7.5t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -7.5t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -7.5t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -7.5t + t2 = 0 + 1.25 -7.5t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -7.5t + t2 = 1.25 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 = 1.25 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = 1.25 + 14.0625 Combine like terms: 1.25 + 14.0625 = 15.3125 14.0625 + -7.5t + t2 = 15.3125 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 15.3125 Calculate the square root of the right side: 3.913118961 Break this problem into two subproblems by setting (t + -3.75) equal to 3.913118961 and -3.913118961.Subproblem 1
t + -3.75 = 3.913118961 Simplifying t + -3.75 = 3.913118961 Reorder the terms: -3.75 + t = 3.913118961 Solving -3.75 + t = 3.913118961 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 = 3.913118961 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 3.913118961 + 3.75 t = 3.913118961 + 3.75 Combine like terms: 3.913118961 + 3.75 = 7.663118961 t = 7.663118961 Simplifying t = 7.663118961Subproblem 2
t + -3.75 = -3.913118961 Simplifying t + -3.75 = -3.913118961 Reorder the terms: -3.75 + t = -3.913118961 Solving -3.75 + t = -3.913118961 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 = -3.913118961 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -3.913118961 + 3.75 t = -3.913118961 + 3.75 Combine like terms: -3.913118961 + 3.75 = -0.163118961 t = -0.163118961 Simplifying t = -0.163118961Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.663118961, -0.163118961}Solution
t = {7.663118961, -0.163118961}
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