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Simplifying 0 = -16t2 + 324t + 42 Reorder the terms: 0 = 42 + 324t + -16t2 Solving 0 = 42 + 324t + -16t2 Solving for variable 't'. Combine like terms: 0 + -42 = -42 -42 + -324t + 16t2 = 42 + 324t + -16t2 + -42 + -324t + 16t2 Reorder the terms: -42 + -324t + 16t2 = 42 + -42 + 324t + -324t + -16t2 + 16t2 Combine like terms: 42 + -42 = 0 -42 + -324t + 16t2 = 0 + 324t + -324t + -16t2 + 16t2 -42 + -324t + 16t2 = 324t + -324t + -16t2 + 16t2 Combine like terms: 324t + -324t = 0 -42 + -324t + 16t2 = 0 + -16t2 + 16t2 -42 + -324t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -42 + -324t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-21 + -162t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-21 + -162t + 8t2)' equal to zero and attempt to solve: Simplifying -21 + -162t + 8t2 = 0 Solving -21 + -162t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.625 + -20.25t + t2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + -20.25t + 2.625 + t2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + -20.25t + t2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + -20.25t + t2 = 0 + 2.625 -20.25t + t2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 -20.25t + t2 = 2.625 The t term is -20.25t. Take half its coefficient (-10.125). Square it (102.515625) and add it to both sides. Add '102.515625' to each side of the equation. -20.25t + 102.515625 + t2 = 2.625 + 102.515625 Reorder the terms: 102.515625 + -20.25t + t2 = 2.625 + 102.515625 Combine like terms: 2.625 + 102.515625 = 105.140625 102.515625 + -20.25t + t2 = 105.140625 Factor a perfect square on the left side: (t + -10.125)(t + -10.125) = 105.140625 Calculate the square root of the right side: 10.253810267 Break this problem into two subproblems by setting (t + -10.125) equal to 10.253810267 and -10.253810267.Subproblem 1
t + -10.125 = 10.253810267 Simplifying t + -10.125 = 10.253810267 Reorder the terms: -10.125 + t = 10.253810267 Solving -10.125 + t = 10.253810267 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10.125' to each side of the equation. -10.125 + 10.125 + t = 10.253810267 + 10.125 Combine like terms: -10.125 + 10.125 = 0.000 0.000 + t = 10.253810267 + 10.125 t = 10.253810267 + 10.125 Combine like terms: 10.253810267 + 10.125 = 20.378810267 t = 20.378810267 Simplifying t = 20.378810267Subproblem 2
t + -10.125 = -10.253810267 Simplifying t + -10.125 = -10.253810267 Reorder the terms: -10.125 + t = -10.253810267 Solving -10.125 + t = -10.253810267 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10.125' to each side of the equation. -10.125 + 10.125 + t = -10.253810267 + 10.125 Combine like terms: -10.125 + 10.125 = 0.000 0.000 + t = -10.253810267 + 10.125 t = -10.253810267 + 10.125 Combine like terms: -10.253810267 + 10.125 = -0.128810267 t = -0.128810267 Simplifying t = -0.128810267Solution
The solution to the problem is based on the solutions from the subproblems. t = {20.378810267, -0.128810267}Solution
t = {20.378810267, -0.128810267}
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