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Simplifying 0 = -16t2 + -22t + 120 Reorder the terms: 0 = 120 + -22t + -16t2 Solving 0 = 120 + -22t + -16t2 Solving for variable 't'. Combine like terms: 0 + -120 = -120 -120 + 22t + 16t2 = 120 + -22t + -16t2 + -120 + 22t + 16t2 Reorder the terms: -120 + 22t + 16t2 = 120 + -120 + -22t + 22t + -16t2 + 16t2 Combine like terms: 120 + -120 = 0 -120 + 22t + 16t2 = 0 + -22t + 22t + -16t2 + 16t2 -120 + 22t + 16t2 = -22t + 22t + -16t2 + 16t2 Combine like terms: -22t + 22t = 0 -120 + 22t + 16t2 = 0 + -16t2 + 16t2 -120 + 22t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -120 + 22t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-60 + 11t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-60 + 11t + 8t2)' equal to zero and attempt to solve: Simplifying -60 + 11t + 8t2 = 0 Solving -60 + 11t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -7.5 + 1.375t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 1.375t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 1.375t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 1.375t + t2 = 0 + 7.5 1.375t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 1.375t + t2 = 7.5 The t term is 1.375t. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375t + 0.47265625 + t2 = 7.5 + 0.47265625 Reorder the terms: 0.47265625 + 1.375t + t2 = 7.5 + 0.47265625 Combine like terms: 7.5 + 0.47265625 = 7.97265625 0.47265625 + 1.375t + t2 = 7.97265625 Factor a perfect square on the left side: (t + 0.6875)(t + 0.6875) = 7.97265625 Calculate the square root of the right side: 2.82358925 Break this problem into two subproblems by setting (t + 0.6875) equal to 2.82358925 and -2.82358925.Subproblem 1
t + 0.6875 = 2.82358925 Simplifying t + 0.6875 = 2.82358925 Reorder the terms: 0.6875 + t = 2.82358925 Solving 0.6875 + t = 2.82358925 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + t = 2.82358925 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = 2.82358925 + -0.6875 t = 2.82358925 + -0.6875 Combine like terms: 2.82358925 + -0.6875 = 2.13608925 t = 2.13608925 Simplifying t = 2.13608925Subproblem 2
t + 0.6875 = -2.82358925 Simplifying t + 0.6875 = -2.82358925 Reorder the terms: 0.6875 + t = -2.82358925 Solving 0.6875 + t = -2.82358925 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + t = -2.82358925 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + t = -2.82358925 + -0.6875 t = -2.82358925 + -0.6875 Combine like terms: -2.82358925 + -0.6875 = -3.51108925 t = -3.51108925 Simplifying t = -3.51108925Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.13608925, -3.51108925}Solution
t = {2.13608925, -3.51108925}
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