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Simplifying 0 = -16t2 + 108t + 6 Reorder the terms: 0 = 6 + 108t + -16t2 Solving 0 = 6 + 108t + -16t2 Solving for variable 't'. Combine like terms: 0 + -6 = -6 -6 + -108t + 16t2 = 6 + 108t + -16t2 + -6 + -108t + 16t2 Reorder the terms: -6 + -108t + 16t2 = 6 + -6 + 108t + -108t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -6 + -108t + 16t2 = 0 + 108t + -108t + -16t2 + 16t2 -6 + -108t + 16t2 = 108t + -108t + -16t2 + 16t2 Combine like terms: 108t + -108t = 0 -6 + -108t + 16t2 = 0 + -16t2 + 16t2 -6 + -108t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6 + -108t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -54t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -54t + 8t2)' equal to zero and attempt to solve: Simplifying -3 + -54t + 8t2 = 0 Solving -3 + -54t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -6.75t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -6.75t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -6.75t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -6.75t + t2 = 0 + 0.375 -6.75t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -6.75t + t2 = 0.375 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 = 0.375 + 11.390625 Reorder the terms: 11.390625 + -6.75t + t2 = 0.375 + 11.390625 Combine like terms: 0.375 + 11.390625 = 11.765625 11.390625 + -6.75t + t2 = 11.765625 Factor a perfect square on the left side: (t + -3.375)(t + -3.375) = 11.765625 Calculate the square root of the right side: 3.430105684 Break this problem into two subproblems by setting (t + -3.375) equal to 3.430105684 and -3.430105684.Subproblem 1
t + -3.375 = 3.430105684 Simplifying t + -3.375 = 3.430105684 Reorder the terms: -3.375 + t = 3.430105684 Solving -3.375 + t = 3.430105684 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 = 3.430105684 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + t = 3.430105684 + 3.375 t = 3.430105684 + 3.375 Combine like terms: 3.430105684 + 3.375 = 6.805105684 t = 6.805105684 Simplifying t = 6.805105684Subproblem 2
t + -3.375 = -3.430105684 Simplifying t + -3.375 = -3.430105684 Reorder the terms: -3.375 + t = -3.430105684 Solving -3.375 + t = -3.430105684 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 = -3.430105684 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + t = -3.430105684 + 3.375 t = -3.430105684 + 3.375 Combine like terms: -3.430105684 + 3.375 = -0.055105684 t = -0.055105684 Simplifying t = -0.055105684Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.805105684, -0.055105684}Solution
t = {6.805105684, -0.055105684}
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