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Simplifying 81 = -16t2 + 80t + 3 Reorder the terms: 81 = 3 + 80t + -16t2 Solving 81 = 3 + 80t + -16t2 Solving for variable 't'. Combine like terms: 81 + -3 = 78 78 + -80t + 16t2 = 3 + 80t + -16t2 + -3 + -80t + 16t2 Reorder the terms: 78 + -80t + 16t2 = 3 + -3 + 80t + -80t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 78 + -80t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 78 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 78 + -80t + 16t2 = 0 + -16t2 + 16t2 78 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 78 + -80t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(39 + -40t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(39 + -40t + 8t2)' equal to zero and attempt to solve: Simplifying 39 + -40t + 8t2 = 0 Solving 39 + -40t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 4.875 + -5t + t2 = 0 Move the constant term to the right: Add '-4.875' to each side of the equation. 4.875 + -5t + -4.875 + t2 = 0 + -4.875 Reorder the terms: 4.875 + -4.875 + -5t + t2 = 0 + -4.875 Combine like terms: 4.875 + -4.875 = 0.000 0.000 + -5t + t2 = 0 + -4.875 -5t + t2 = 0 + -4.875 Combine like terms: 0 + -4.875 = -4.875 -5t + t2 = -4.875 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 = -4.875 + 6.25 Reorder the terms: 6.25 + -5t + t2 = -4.875 + 6.25 Combine like terms: -4.875 + 6.25 = 1.375 6.25 + -5t + t2 = 1.375 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 1.375 Calculate the square root of the right side: 1.17260394 Break this problem into two subproblems by setting (t + -2.5) equal to 1.17260394 and -1.17260394.Subproblem 1
t + -2.5 = 1.17260394 Simplifying t + -2.5 = 1.17260394 Reorder the terms: -2.5 + t = 1.17260394 Solving -2.5 + t = 1.17260394 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 = 1.17260394 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 1.17260394 + 2.5 t = 1.17260394 + 2.5 Combine like terms: 1.17260394 + 2.5 = 3.67260394 t = 3.67260394 Simplifying t = 3.67260394Subproblem 2
t + -2.5 = -1.17260394 Simplifying t + -2.5 = -1.17260394 Reorder the terms: -2.5 + t = -1.17260394 Solving -2.5 + t = -1.17260394 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 = -1.17260394 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -1.17260394 + 2.5 t = -1.17260394 + 2.5 Combine like terms: -1.17260394 + 2.5 = 1.32739606 t = 1.32739606 Simplifying t = 1.32739606Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.67260394, 1.32739606}Solution
t = {3.67260394, 1.32739606}
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