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Simplifying 8 = -16t2 + 96t + 116 Reorder the terms: 8 = 116 + 96t + -16t2 Solving 8 = 116 + 96t + -16t2 Solving for variable 't'. Combine like terms: 8 + -116 = -108 -108 + -96t + 16t2 = 116 + 96t + -16t2 + -116 + -96t + 16t2 Reorder the terms: -108 + -96t + 16t2 = 116 + -116 + 96t + -96t + -16t2 + 16t2 Combine like terms: 116 + -116 = 0 -108 + -96t + 16t2 = 0 + 96t + -96t + -16t2 + 16t2 -108 + -96t + 16t2 = 96t + -96t + -16t2 + 16t2 Combine like terms: 96t + -96t = 0 -108 + -96t + 16t2 = 0 + -16t2 + 16t2 -108 + -96t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -108 + -96t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-27 + -24t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-27 + -24t + 4t2)' equal to zero and attempt to solve: Simplifying -27 + -24t + 4t2 = 0 Solving -27 + -24t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.75 + -6t + t2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + -6t + 6.75 + t2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + -6t + t2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + -6t + t2 = 0 + 6.75 -6t + t2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 -6t + t2 = 6.75 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = 6.75 + 9 Reorder the terms: 9 + -6t + t2 = 6.75 + 9 Combine like terms: 6.75 + 9 = 15.75 9 + -6t + t2 = 15.75 Factor a perfect square on the left side: (t + -3)(t + -3) = 15.75 Calculate the square root of the right side: 3.968626967 Break this problem into two subproblems by setting (t + -3) equal to 3.968626967 and -3.968626967.Subproblem 1
t + -3 = 3.968626967 Simplifying t + -3 = 3.968626967 Reorder the terms: -3 + t = 3.968626967 Solving -3 + t = 3.968626967 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 3.968626967 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.968626967 + 3 t = 3.968626967 + 3 Combine like terms: 3.968626967 + 3 = 6.968626967 t = 6.968626967 Simplifying t = 6.968626967Subproblem 2
t + -3 = -3.968626967 Simplifying t + -3 = -3.968626967 Reorder the terms: -3 + t = -3.968626967 Solving -3 + t = -3.968626967 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -3.968626967 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.968626967 + 3 t = -3.968626967 + 3 Combine like terms: -3.968626967 + 3 = -0.968626967 t = -0.968626967 Simplifying t = -0.968626967Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.968626967, -0.968626967}Solution
t = {6.968626967, -0.968626967}
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