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Simplifying 0 = -16t2 + 32t + 324 Reorder the terms: 0 = 324 + 32t + -16t2 Solving 0 = 324 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0 + -324 = -324 -324 + -32t + 16t2 = 324 + 32t + -16t2 + -324 + -32t + 16t2 Reorder the terms: -324 + -32t + 16t2 = 324 + -324 + 32t + -32t + -16t2 + 16t2 Combine like terms: 324 + -324 = 0 -324 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -324 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -324 + -32t + 16t2 = 0 + -16t2 + 16t2 -324 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -324 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-81 + -8t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-81 + -8t + 4t2)' equal to zero and attempt to solve: Simplifying -81 + -8t + 4t2 = 0 Solving -81 + -8t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -20.25 + -2t + t2 = 0 Move the constant term to the right: Add '20.25' to each side of the equation. -20.25 + -2t + 20.25 + t2 = 0 + 20.25 Reorder the terms: -20.25 + 20.25 + -2t + t2 = 0 + 20.25 Combine like terms: -20.25 + 20.25 = 0.00 0.00 + -2t + t2 = 0 + 20.25 -2t + t2 = 0 + 20.25 Combine like terms: 0 + 20.25 = 20.25 -2t + t2 = 20.25 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 = 20.25 + 1 Reorder the terms: 1 + -2t + t2 = 20.25 + 1 Combine like terms: 20.25 + 1 = 21.25 1 + -2t + t2 = 21.25 Factor a perfect square on the left side: (t + -1)(t + -1) = 21.25 Calculate the square root of the right side: 4.609772229 Break this problem into two subproblems by setting (t + -1) equal to 4.609772229 and -4.609772229.Subproblem 1
t + -1 = 4.609772229 Simplifying t + -1 = 4.609772229 Reorder the terms: -1 + t = 4.609772229 Solving -1 + t = 4.609772229 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 = 4.609772229 + 1 Combine like terms: -1 + 1 = 0 0 + t = 4.609772229 + 1 t = 4.609772229 + 1 Combine like terms: 4.609772229 + 1 = 5.609772229 t = 5.609772229 Simplifying t = 5.609772229Subproblem 2
t + -1 = -4.609772229 Simplifying t + -1 = -4.609772229 Reorder the terms: -1 + t = -4.609772229 Solving -1 + t = -4.609772229 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 = -4.609772229 + 1 Combine like terms: -1 + 1 = 0 0 + t = -4.609772229 + 1 t = -4.609772229 + 1 Combine like terms: -4.609772229 + 1 = -3.609772229 t = -3.609772229 Simplifying t = -3.609772229Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.609772229, -3.609772229}Solution
t = {5.609772229, -3.609772229}
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