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Simplifying 20(t) = -16t2 + -32t + 5 Reorder the terms: 20t = 5 + -32t + -16t2 Solving 20t = 5 + -32t + -16t2 Solving for variable 't'. Reorder the terms: -5 + 20t + 32t + 16t2 = 5 + -32t + -16t2 + -5 + 32t + 16t2 Combine like terms: 20t + 32t = 52t -5 + 52t + 16t2 = 5 + -32t + -16t2 + -5 + 32t + 16t2 Reorder the terms: -5 + 52t + 16t2 = 5 + -5 + -32t + 32t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -5 + 52t + 16t2 = 0 + -32t + 32t + -16t2 + 16t2 -5 + 52t + 16t2 = -32t + 32t + -16t2 + 16t2 Combine like terms: -32t + 32t = 0 -5 + 52t + 16t2 = 0 + -16t2 + 16t2 -5 + 52t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -5 + 52t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + 3.25t + t2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + 3.25t + 0.3125 + t2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + 3.25t + t2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + 3.25t + t2 = 0 + 0.3125 3.25t + t2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 3.25t + t2 = 0.3125 The t term is 3.25t. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25t + 2.640625 + t2 = 0.3125 + 2.640625 Reorder the terms: 2.640625 + 3.25t + t2 = 0.3125 + 2.640625 Combine like terms: 0.3125 + 2.640625 = 2.953125 2.640625 + 3.25t + t2 = 2.953125 Factor a perfect square on the left side: (t + 1.625)(t + 1.625) = 2.953125 Calculate the square root of the right side: 1.718465886 Break this problem into two subproblems by setting (t + 1.625) equal to 1.718465886 and -1.718465886.Subproblem 1
t + 1.625 = 1.718465886 Simplifying t + 1.625 = 1.718465886 Reorder the terms: 1.625 + t = 1.718465886 Solving 1.625 + t = 1.718465886 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + t = 1.718465886 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + t = 1.718465886 + -1.625 t = 1.718465886 + -1.625 Combine like terms: 1.718465886 + -1.625 = 0.093465886 t = 0.093465886 Simplifying t = 0.093465886Subproblem 2
t + 1.625 = -1.718465886 Simplifying t + 1.625 = -1.718465886 Reorder the terms: 1.625 + t = -1.718465886 Solving 1.625 + t = -1.718465886 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + t = -1.718465886 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + t = -1.718465886 + -1.625 t = -1.718465886 + -1.625 Combine like terms: -1.718465886 + -1.625 = -3.343465886 t = -3.343465886 Simplifying t = -3.343465886Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.093465886, -3.343465886}
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