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Simplifying 16t2 + 96t + -106 = 0 Reorder the terms: -106 + 96t + 16t2 = 0 Solving -106 + 96t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(-53 + 48t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-53 + 48t + 8t2)' equal to zero and attempt to solve: Simplifying -53 + 48t + 8t2 = 0 Solving -53 + 48t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -6.625 + 6t + t2 = 0 Move the constant term to the right: Add '6.625' to each side of the equation. -6.625 + 6t + 6.625 + t2 = 0 + 6.625 Reorder the terms: -6.625 + 6.625 + 6t + t2 = 0 + 6.625 Combine like terms: -6.625 + 6.625 = 0.000 0.000 + 6t + t2 = 0 + 6.625 6t + t2 = 0 + 6.625 Combine like terms: 0 + 6.625 = 6.625 6t + t2 = 6.625 The t term is 6t. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6t + 9 + t2 = 6.625 + 9 Reorder the terms: 9 + 6t + t2 = 6.625 + 9 Combine like terms: 6.625 + 9 = 15.625 9 + 6t + t2 = 15.625 Factor a perfect square on the left side: (t + 3)(t + 3) = 15.625 Calculate the square root of the right side: 3.952847075 Break this problem into two subproblems by setting (t + 3) equal to 3.952847075 and -3.952847075.Subproblem 1
t + 3 = 3.952847075 Simplifying t + 3 = 3.952847075 Reorder the terms: 3 + t = 3.952847075 Solving 3 + t = 3.952847075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = 3.952847075 + -3 Combine like terms: 3 + -3 = 0 0 + t = 3.952847075 + -3 t = 3.952847075 + -3 Combine like terms: 3.952847075 + -3 = 0.952847075 t = 0.952847075 Simplifying t = 0.952847075Subproblem 2
t + 3 = -3.952847075 Simplifying t + 3 = -3.952847075 Reorder the terms: 3 + t = -3.952847075 Solving 3 + t = -3.952847075 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = -3.952847075 + -3 Combine like terms: 3 + -3 = 0 0 + t = -3.952847075 + -3 t = -3.952847075 + -3 Combine like terms: -3.952847075 + -3 = -6.952847075 t = -6.952847075 Simplifying t = -6.952847075Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.952847075, -6.952847075}Solution
t = {0.952847075, -6.952847075}
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