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Simplifying 7 = -16t2 + 80t Reorder the terms: 7 = 80t + -16t2 Solving 7 = 80t + -16t2 Solving for variable 't'. Reorder the terms: 7 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 7 + -80t + 16t2 = 0 + -16t2 + 16t2 7 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 7 + -80t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + -5t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + -5t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + -5t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + -5t + t2 = 0 + -0.4375 -5t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 -5t + t2 = -0.4375 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = -0.4375 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -0.4375 + 6.25 Combine like terms: -0.4375 + 6.25 = 5.8125 6.25 + -5t + t2 = 5.8125 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 5.8125 Calculate the square root of the right side: 2.41091269 Break this problem into two subproblems by setting (t + -2.5) equal to 2.41091269 and -2.41091269.Subproblem 1
t + -2.5 = 2.41091269 Simplifying t + -2.5 = 2.41091269 Reorder the terms: -2.5 + t = 2.41091269 Solving -2.5 + t = 2.41091269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 2.41091269 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.41091269 + 2.5 t = 2.41091269 + 2.5 Combine like terms: 2.41091269 + 2.5 = 4.91091269 t = 4.91091269 Simplifying t = 4.91091269Subproblem 2
t + -2.5 = -2.41091269 Simplifying t + -2.5 = -2.41091269 Reorder the terms: -2.5 + t = -2.41091269 Solving -2.5 + t = -2.41091269 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -2.41091269 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.41091269 + 2.5 t = -2.41091269 + 2.5 Combine like terms: -2.41091269 + 2.5 = 0.08908731 t = 0.08908731 Simplifying t = 0.08908731Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.91091269, 0.08908731}
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