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Simplifying 20 = -16t2 + 120t + 5 Reorder the terms: 20 = 5 + 120t + -16t2 Solving 20 = 5 + 120t + -16t2 Solving for variable 't'. Combine like terms: 20 + -5 = 15 15 + -120t + 16t2 = 5 + 120t + -16t2 + -5 + -120t + 16t2 Reorder the terms: 15 + -120t + 16t2 = 5 + -5 + 120t + -120t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 15 + -120t + 16t2 = 0 + 120t + -120t + -16t2 + 16t2 15 + -120t + 16t2 = 120t + -120t + -16t2 + 16t2 Combine like terms: 120t + -120t = 0 15 + -120t + 16t2 = 0 + -16t2 + 16t2 15 + -120t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 15 + -120t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.9375 + -7.5t + t2 = 0 Move the constant term to the right: Add '-0.9375' to each side of the equation. 0.9375 + -7.5t + -0.9375 + t2 = 0 + -0.9375 Reorder the terms: 0.9375 + -0.9375 + -7.5t + t2 = 0 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + -7.5t + t2 = 0 + -0.9375 -7.5t + t2 = 0 + -0.9375 Combine like terms: 0 + -0.9375 = -0.9375 -7.5t + t2 = -0.9375 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 = -0.9375 + 14.0625 Reorder the terms: 14.0625 + -7.5t + t2 = -0.9375 + 14.0625 Combine like terms: -0.9375 + 14.0625 = 13.125 14.0625 + -7.5t + t2 = 13.125 Factor a perfect square on the left side: (t + -3.75)(t + -3.75) = 13.125 Calculate the square root of the right side: 3.622844187 Break this problem into two subproblems by setting (t + -3.75) equal to 3.622844187 and -3.622844187.Subproblem 1
t + -3.75 = 3.622844187 Simplifying t + -3.75 = 3.622844187 Reorder the terms: -3.75 + t = 3.622844187 Solving -3.75 + t = 3.622844187 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 = 3.622844187 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = 3.622844187 + 3.75 t = 3.622844187 + 3.75 Combine like terms: 3.622844187 + 3.75 = 7.372844187 t = 7.372844187 Simplifying t = 7.372844187Subproblem 2
t + -3.75 = -3.622844187 Simplifying t + -3.75 = -3.622844187 Reorder the terms: -3.75 + t = -3.622844187 Solving -3.75 + t = -3.622844187 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 = -3.622844187 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + t = -3.622844187 + 3.75 t = -3.622844187 + 3.75 Combine like terms: -3.622844187 + 3.75 = 0.127155813 t = 0.127155813 Simplifying t = 0.127155813Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.372844187, 0.127155813}
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