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Simplifying 0 = -16t2 + 32t + -8 Reorder the terms: 0 = -8 + 32t + -16t2 Solving 0 = -8 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0 + 8 = 8 8 + -32t + 16t2 = -8 + 32t + -16t2 + 8 + -32t + 16t2 Reorder the terms: 8 + -32t + 16t2 = -8 + 8 + 32t + -32t + -16t2 + 16t2 Combine like terms: -8 + 8 = 0 8 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 8 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 8 + -32t + 16t2 = 0 + -16t2 + 16t2 8 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 8 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(1 + -4t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(1 + -4t + 2t2)' equal to zero and attempt to solve: Simplifying 1 + -4t + 2t2 = 0 Solving 1 + -4t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2t + t2 = 0 + -0.5 -2t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2t + t2 = -0.5 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = -0.5 + 1 Reorder the terms: 1 + -2t + t2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2t + t2 = 0.5 Factor a perfect square on the left side: (t + -1)(t + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (t + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
t + -1 = 0.707106781 Simplifying t + -1 = 0.707106781 Reorder the terms: -1 + t = 0.707106781 Solving -1 + t = 0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + t = 0.707106781 + 1 t = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 t = 1.707106781 Simplifying t = 1.707106781Subproblem 2
t + -1 = -0.707106781 Simplifying t + -1 = -0.707106781 Reorder the terms: -1 + t = -0.707106781 Solving -1 + t = -0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + t = -0.707106781 + 1 t = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 t = 0.292893219 Simplifying t = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.707106781, 0.292893219}Solution
t = {1.707106781, 0.292893219}
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