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Simplifying 0 = -16t2 + 108t + 1068 Reorder the terms: 0 = 1068 + 108t + -16t2 Solving 0 = 1068 + 108t + -16t2 Solving for variable 't'. Combine like terms: 0 + -1068 = -1068 -1068 + -108t + 16t2 = 1068 + 108t + -16t2 + -1068 + -108t + 16t2 Reorder the terms: -1068 + -108t + 16t2 = 1068 + -1068 + 108t + -108t + -16t2 + 16t2 Combine like terms: 1068 + -1068 = 0 -1068 + -108t + 16t2 = 0 + 108t + -108t + -16t2 + 16t2 -1068 + -108t + 16t2 = 108t + -108t + -16t2 + 16t2 Combine like terms: 108t + -108t = 0 -1068 + -108t + 16t2 = 0 + -16t2 + 16t2 -1068 + -108t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -1068 + -108t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-267 + -27t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-267 + -27t + 4t2)' equal to zero and attempt to solve: Simplifying -267 + -27t + 4t2 = 0 Solving -267 + -27t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -66.75 + -6.75t + t2 = 0 Move the constant term to the right: Add '66.75' to each side of the equation. -66.75 + -6.75t + 66.75 + t2 = 0 + 66.75 Reorder the terms: -66.75 + 66.75 + -6.75t + t2 = 0 + 66.75 Combine like terms: -66.75 + 66.75 = 0.00 0.00 + -6.75t + t2 = 0 + 66.75 -6.75t + t2 = 0 + 66.75 Combine like terms: 0 + 66.75 = 66.75 -6.75t + t2 = 66.75 The t term is -6.75t. Take half its coefficient (-3.375). Square it (11.390625) and add it to both sides. Add '11.390625' to each side of the equation. -6.75t + 11.390625 + t2 = 66.75 + 11.390625 Reorder the terms: 11.390625 + -6.75t + t2 = 66.75 + 11.390625 Combine like terms: 66.75 + 11.390625 = 78.140625 11.390625 + -6.75t + t2 = 78.140625 Factor a perfect square on the left side: (t + -3.375)(t + -3.375) = 78.140625 Calculate the square root of the right side: 8.839718604 Break this problem into two subproblems by setting (t + -3.375) equal to 8.839718604 and -8.839718604.Subproblem 1
t + -3.375 = 8.839718604 Simplifying t + -3.375 = 8.839718604 Reorder the terms: -3.375 + t = 8.839718604 Solving -3.375 + t = 8.839718604 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + t = 8.839718604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + t = 8.839718604 + 3.375 t = 8.839718604 + 3.375 Combine like terms: 8.839718604 + 3.375 = 12.214718604 t = 12.214718604 Simplifying t = 12.214718604Subproblem 2
t + -3.375 = -8.839718604 Simplifying t + -3.375 = -8.839718604 Reorder the terms: -3.375 + t = -8.839718604 Solving -3.375 + t = -8.839718604 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + t = -8.839718604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + t = -8.839718604 + 3.375 t = -8.839718604 + 3.375 Combine like terms: -8.839718604 + 3.375 = -5.464718604 t = -5.464718604 Simplifying t = -5.464718604Solution
The solution to the problem is based on the solutions from the subproblems. t = {12.214718604, -5.464718604}Solution
t = {12.214718604, -5.464718604}
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