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Simplifying 0 = (16t2 + -72t + -81) Reorder the terms: 0 = (-81 + -72t + 16t2) Remove parenthesis around (-81 + -72t + 16t2) 0 = -81 + -72t + 16t2 Solving 0 = -81 + -72t + 16t2 Solving for variable 't'. Combine like terms: 0 + 81 = 81 81 + 72t + -16t2 = -81 + -72t + 16t2 + 81 + 72t + -16t2 Reorder the terms: 81 + 72t + -16t2 = -81 + 81 + -72t + 72t + 16t2 + -16t2 Combine like terms: -81 + 81 = 0 81 + 72t + -16t2 = 0 + -72t + 72t + 16t2 + -16t2 81 + 72t + -16t2 = -72t + 72t + 16t2 + -16t2 Combine like terms: -72t + 72t = 0 81 + 72t + -16t2 = 0 + 16t2 + -16t2 81 + 72t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 81 + 72t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -5.0625 + -4.5t + t2 = 0 Move the constant term to the right: Add '5.0625' to each side of the equation. -5.0625 + -4.5t + 5.0625 + t2 = 0 + 5.0625 Reorder the terms: -5.0625 + 5.0625 + -4.5t + t2 = 0 + 5.0625 Combine like terms: -5.0625 + 5.0625 = 0.0000 0.0000 + -4.5t + t2 = 0 + 5.0625 -4.5t + t2 = 0 + 5.0625 Combine like terms: 0 + 5.0625 = 5.0625 -4.5t + t2 = 5.0625 The t term is -4.5t. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5t + 5.0625 + t2 = 5.0625 + 5.0625 Reorder the terms: 5.0625 + -4.5t + t2 = 5.0625 + 5.0625 Combine like terms: 5.0625 + 5.0625 = 10.125 5.0625 + -4.5t + t2 = 10.125 Factor a perfect square on the left side: (t + -2.25)(t + -2.25) = 10.125 Calculate the square root of the right side: 3.181980515 Break this problem into two subproblems by setting (t + -2.25) equal to 3.181980515 and -3.181980515.Subproblem 1
t + -2.25 = 3.181980515 Simplifying t + -2.25 = 3.181980515 Reorder the terms: -2.25 + t = 3.181980515 Solving -2.25 + t = 3.181980515 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = 3.181980515 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = 3.181980515 + 2.25 t = 3.181980515 + 2.25 Combine like terms: 3.181980515 + 2.25 = 5.431980515 t = 5.431980515 Simplifying t = 5.431980515Subproblem 2
t + -2.25 = -3.181980515 Simplifying t + -2.25 = -3.181980515 Reorder the terms: -2.25 + t = -3.181980515 Solving -2.25 + t = -3.181980515 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = -3.181980515 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = -3.181980515 + 2.25 t = -3.181980515 + 2.25 Combine like terms: -3.181980515 + 2.25 = -0.931980515 t = -0.931980515 Simplifying t = -0.931980515Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.431980515, -0.931980515}
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