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Simplifying 16t2 + 32t + -206 = 0 Reorder the terms: -206 + 32t + 16t2 = 0 Solving -206 + 32t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-103 + 16t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-103 + 16t + 8t2)' equal to zero and attempt to solve: Simplifying -103 + 16t + 8t2 = 0 Solving -103 + 16t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -12.875 + 2t + t2 = 0 Move the constant term to the right: Add '12.875' to each side of the equation. -12.875 + 2t + 12.875 + t2 = 0 + 12.875 Reorder the terms: -12.875 + 12.875 + 2t + t2 = 0 + 12.875 Combine like terms: -12.875 + 12.875 = 0.000 0.000 + 2t + t2 = 0 + 12.875 2t + t2 = 0 + 12.875 Combine like terms: 0 + 12.875 = 12.875 2t + t2 = 12.875 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 = 12.875 + 1 Reorder the terms: 1 + 2t + t2 = 12.875 + 1 Combine like terms: 12.875 + 1 = 13.875 1 + 2t + t2 = 13.875 Factor a perfect square on the left side: (t + 1)(t + 1) = 13.875 Calculate the square root of the right side: 3.724916106 Break this problem into two subproblems by setting (t + 1) equal to 3.724916106 and -3.724916106.Subproblem 1
t + 1 = 3.724916106 Simplifying t + 1 = 3.724916106 Reorder the terms: 1 + t = 3.724916106 Solving 1 + t = 3.724916106 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 = 3.724916106 + -1 Combine like terms: 1 + -1 = 0 0 + t = 3.724916106 + -1 t = 3.724916106 + -1 Combine like terms: 3.724916106 + -1 = 2.724916106 t = 2.724916106 Simplifying t = 2.724916106Subproblem 2
t + 1 = -3.724916106 Simplifying t + 1 = -3.724916106 Reorder the terms: 1 + t = -3.724916106 Solving 1 + t = -3.724916106 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 = -3.724916106 + -1 Combine like terms: 1 + -1 = 0 0 + t = -3.724916106 + -1 t = -3.724916106 + -1 Combine like terms: -3.724916106 + -1 = -4.724916106 t = -4.724916106 Simplifying t = -4.724916106Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.724916106, -4.724916106}Solution
t = {2.724916106, -4.724916106}
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