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Simplifying 2 = -16t2 + 64t Reorder the terms: 2 = 64t + -16t2 Solving 2 = 64t + -16t2 Solving for variable 't'. Reorder the terms: 2 + -64t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 2 + -64t + 16t2 = 0 + -16t2 + 16t2 2 + -64t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 2 + -64t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + -32t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + -32t + 8t2)' equal to zero and attempt to solve: Simplifying 1 + -32t + 8t2 = 0 Solving 1 + -32t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + -4t + t2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -4t + -0.125 + t2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -4t + t2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -4t + t2 = 0 + -0.125 -4t + t2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -4t + t2 = -0.125 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -0.125 + 4 Reorder the terms: 4 + -4t + t2 = -0.125 + 4 Combine like terms: -0.125 + 4 = 3.875 4 + -4t + t2 = 3.875 Factor a perfect square on the left side: (t + -2)(t + -2) = 3.875 Calculate the square root of the right side: 1.968501969 Break this problem into two subproblems by setting (t + -2) equal to 1.968501969 and -1.968501969.Subproblem 1
t + -2 = 1.968501969 Simplifying t + -2 = 1.968501969 Reorder the terms: -2 + t = 1.968501969 Solving -2 + t = 1.968501969 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.968501969 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.968501969 + 2 t = 1.968501969 + 2 Combine like terms: 1.968501969 + 2 = 3.968501969 t = 3.968501969 Simplifying t = 3.968501969Subproblem 2
t + -2 = -1.968501969 Simplifying t + -2 = -1.968501969 Reorder the terms: -2 + t = -1.968501969 Solving -2 + t = -1.968501969 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.968501969 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.968501969 + 2 t = -1.968501969 + 2 Combine like terms: -1.968501969 + 2 = 0.031498031 t = 0.031498031 Simplifying t = 0.031498031Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.968501969, 0.031498031}Solution
t = {3.968501969, 0.031498031}
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