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Simplifying 16t2 + -122t + 192 = 0 Reorder the terms: 192 + -122t + 16t2 = 0 Solving 192 + -122t + 16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(96 + -61t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(96 + -61t + 8t2)' equal to zero and attempt to solve: Simplifying 96 + -61t + 8t2 = 0 Solving 96 + -61t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 12 + -7.625t + t2 = 0 Move the constant term to the right: Add '-12' to each side of the equation. 12 + -7.625t + -12 + t2 = 0 + -12 Reorder the terms: 12 + -12 + -7.625t + t2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -7.625t + t2 = 0 + -12 -7.625t + t2 = 0 + -12 Combine like terms: 0 + -12 = -12 -7.625t + t2 = -12 The t term is -7.625t. Take half its coefficient (-3.8125). Square it (14.53515625) and add it to both sides. Add '14.53515625' to each side of the equation. -7.625t + 14.53515625 + t2 = -12 + 14.53515625 Reorder the terms: 14.53515625 + -7.625t + t2 = -12 + 14.53515625 Combine like terms: -12 + 14.53515625 = 2.53515625 14.53515625 + -7.625t + t2 = 2.53515625 Factor a perfect square on the left side: (t + -3.8125)(t + -3.8125) = 2.53515625 Calculate the square root of the right side: 1.5922174 Break this problem into two subproblems by setting (t + -3.8125) equal to 1.5922174 and -1.5922174.Subproblem 1
t + -3.8125 = 1.5922174 Simplifying t + -3.8125 = 1.5922174 Reorder the terms: -3.8125 + t = 1.5922174 Solving -3.8125 + t = 1.5922174 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.8125' to each side of the equation. -3.8125 + 3.8125 + t = 1.5922174 + 3.8125 Combine like terms: -3.8125 + 3.8125 = 0.0000 0.0000 + t = 1.5922174 + 3.8125 t = 1.5922174 + 3.8125 Combine like terms: 1.5922174 + 3.8125 = 5.4047174 t = 5.4047174 Simplifying t = 5.4047174Subproblem 2
t + -3.8125 = -1.5922174 Simplifying t + -3.8125 = -1.5922174 Reorder the terms: -3.8125 + t = -1.5922174 Solving -3.8125 + t = -1.5922174 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.8125' to each side of the equation. -3.8125 + 3.8125 + t = -1.5922174 + 3.8125 Combine like terms: -3.8125 + 3.8125 = 0.0000 0.0000 + t = -1.5922174 + 3.8125 t = -1.5922174 + 3.8125 Combine like terms: -1.5922174 + 3.8125 = 2.2202826 t = 2.2202826 Simplifying t = 2.2202826Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.4047174, 2.2202826}Solution
t = {5.4047174, 2.2202826}
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