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Simplifying 120 = -16t2 + 120t + 30 Reorder the terms: 120 = 30 + 120t + -16t2 Solving 120 = 30 + 120t + -16t2 Solving for variable 't'. Combine like terms: 120 + -30 = 90 90 + -120t + 16t2 = 30 + 120t + -16t2 + -30 + -120t + 16t2 Reorder the terms: 90 + -120t + 16t2 = 30 + -30 + 120t + -120t + -16t2 + 16t2 Combine like terms: 30 + -30 = 0 90 + -120t + 16t2 = 0 + 120t + -120t + -16t2 + 16t2 90 + -120t + 16t2 = 120t + -120t + -16t2 + 16t2 Combine like terms: 120t + -120t = 0 90 + -120t + 16t2 = 0 + -16t2 + 16t2 90 + -120t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 90 + -120t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(45 + -60t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(45 + -60t + 8t2)' equal to zero and attempt to solve: Simplifying 45 + -60t + 8t2 = 0 Solving 45 + -60t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 5.625 + -7.5t + t2 = 0 Move the constant term to the right: Add '-5.625' to each side of the equation. 5.625 + -7.5t + -5.625 + t2 = 0 + -5.625 Reorder the terms: 5.625 + -5.625 + -7.5t + t2 = 0 + -5.625 Combine like terms: 5.625 + -5.625 = 0.000 0.000 + -7.5t + t2 = 0 + -5.625 -7.5t + t2 = 0 + -5.625 Combine like terms: 0 + -5.625 = -5.625 -7.5t + t2 = -5.625 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 = -5.625 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -5.625 + 14.0625 Combine like terms: -5.625 + 14.0625 = 8.4375 14.0625 + -7.5t + t2 = 8.4375 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 8.4375 Calculate the square root of the right side: 2.90473751 Break this problem into two subproblems by setting (t + -3.75) equal to 2.90473751 and -2.90473751.Subproblem 1
t + -3.75 = 2.90473751 Simplifying t + -3.75 = 2.90473751 Reorder the terms: -3.75 + t = 2.90473751 Solving -3.75 + t = 2.90473751 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 = 2.90473751 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 2.90473751 + 3.75 t = 2.90473751 + 3.75 Combine like terms: 2.90473751 + 3.75 = 6.65473751 t = 6.65473751 Simplifying t = 6.65473751Subproblem 2
t + -3.75 = -2.90473751 Simplifying t + -3.75 = -2.90473751 Reorder the terms: -3.75 + t = -2.90473751 Solving -3.75 + t = -2.90473751 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 = -2.90473751 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -2.90473751 + 3.75 t = -2.90473751 + 3.75 Combine like terms: -2.90473751 + 3.75 = 0.84526249 t = 0.84526249 Simplifying t = 0.84526249Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.65473751, 0.84526249}Solution
t = {6.65473751, 0.84526249}
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