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Simplifying 30 = -16t2 + 80t + 5 Reorder the terms: 30 = 5 + 80t + -16t2 Solving 30 = 5 + 80t + -16t2 Solving for variable 't'. Combine like terms: 30 + -5 = 25 25 + -80t + 16t2 = 5 + 80t + -16t2 + -5 + -80t + 16t2 Reorder the terms: 25 + -80t + 16t2 = 5 + -5 + 80t + -80t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 25 + -80t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 25 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 25 + -80t + 16t2 = 0 + -16t2 + 16t2 25 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 25 + -80t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.5625 + -5t + t2 = 0 Move the constant term to the right: Add '-1.5625' to each side of the equation. 1.5625 + -5t + -1.5625 + t2 = 0 + -1.5625 Reorder the terms: 1.5625 + -1.5625 + -5t + t2 = 0 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + -5t + t2 = 0 + -1.5625 -5t + t2 = 0 + -1.5625 Combine like terms: 0 + -1.5625 = -1.5625 -5t + t2 = -1.5625 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 = -1.5625 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -1.5625 + 6.25 Combine like terms: -1.5625 + 6.25 = 4.6875 6.25 + -5t + t2 = 4.6875 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 4.6875 Calculate the square root of the right side: 2.165063509 Break this problem into two subproblems by setting (t + -2.5) equal to 2.165063509 and -2.165063509.Subproblem 1
t + -2.5 = 2.165063509 Simplifying t + -2.5 = 2.165063509 Reorder the terms: -2.5 + t = 2.165063509 Solving -2.5 + t = 2.165063509 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 = 2.165063509 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.165063509 + 2.5 t = 2.165063509 + 2.5 Combine like terms: 2.165063509 + 2.5 = 4.665063509 t = 4.665063509 Simplifying t = 4.665063509Subproblem 2
t + -2.5 = -2.165063509 Simplifying t + -2.5 = -2.165063509 Reorder the terms: -2.5 + t = -2.165063509 Solving -2.5 + t = -2.165063509 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 = -2.165063509 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.165063509 + 2.5 t = -2.165063509 + 2.5 Combine like terms: -2.165063509 + 2.5 = 0.334936491 t = 0.334936491 Simplifying t = 0.334936491Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.665063509, 0.334936491}
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