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Simplifying 0 = 16t2 + -32t + 6 Reorder the terms: 0 = 6 + -32t + 16t2 Solving 0 = 6 + -32t + 16t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + 32t + -16t2 = 6 + -32t + 16t2 + -6 + 32t + -16t2 Reorder the terms: -6 + 32t + -16t2 = 6 + -6 + -32t + 32t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 -6 + 32t + -16t2 = 0 + -32t + 32t + 16t2 + -16t2 -6 + 32t + -16t2 = -32t + 32t + 16t2 + -16t2 Combine like terms: -32t + 32t = 0 -6 + 32t + -16t2 = 0 + 16t2 + -16t2 -6 + 32t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -6 + 32t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 16t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 16t + -8t2)' equal to zero and attempt to solve: Simplifying -3 + 16t + -8t2 = 0 Solving -3 + 16t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 0.375 + -2t + t2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -2t + -0.375 + t2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -2t + t2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -2t + t2 = 0 + -0.375 -2t + t2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -2t + t2 = -0.375 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 = -0.375 + 1 Reorder the terms: 1 + -2t + t2 = -0.375 + 1 Combine like terms: -0.375 + 1 = 0.625 1 + -2t + t2 = 0.625 Factor a perfect square on the left side: (t + -1)(t + -1) = 0.625 Calculate the square root of the right side: 0.790569415 Break this problem into two subproblems by setting (t + -1) equal to 0.790569415 and -0.790569415.Subproblem 1
t + -1 = 0.790569415 Simplifying t + -1 = 0.790569415 Reorder the terms: -1 + t = 0.790569415 Solving -1 + t = 0.790569415 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 = 0.790569415 + 1 Combine like terms: -1 + 1 = 0 0 + t = 0.790569415 + 1 t = 0.790569415 + 1 Combine like terms: 0.790569415 + 1 = 1.790569415 t = 1.790569415 Simplifying t = 1.790569415Subproblem 2
t + -1 = -0.790569415 Simplifying t + -1 = -0.790569415 Reorder the terms: -1 + t = -0.790569415 Solving -1 + t = -0.790569415 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 = -0.790569415 + 1 Combine like terms: -1 + 1 = 0 0 + t = -0.790569415 + 1 t = -0.790569415 + 1 Combine like terms: -0.790569415 + 1 = 0.209430585 t = 0.209430585 Simplifying t = 0.209430585Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.790569415, 0.209430585}Solution
t = {1.790569415, 0.209430585}
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