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Simplifying 8t2 + 28t + -108 = 0 Reorder the terms: -108 + 28t + 8t2 = 0 Solving -108 + 28t + 8t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(-27 + 7t + 2t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-27 + 7t + 2t2)' equal to zero and attempt to solve: Simplifying -27 + 7t + 2t2 = 0 Solving -27 + 7t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -13.5 + 3.5t + t2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 3.5t + 13.5 + t2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 3.5t + t2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 3.5t + t2 = 0 + 13.5 3.5t + t2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 3.5t + t2 = 13.5 The t term is 3.5t. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5t + 3.0625 + t2 = 13.5 + 3.0625 Reorder the terms: 3.0625 + 3.5t + t2 = 13.5 + 3.0625 Combine like terms: 13.5 + 3.0625 = 16.5625 3.0625 + 3.5t + t2 = 16.5625 Factor a perfect square on the left side: (t + 1.75)(t + 1.75) = 16.5625 Calculate the square root of the right side: 4.069705149 Break this problem into two subproblems by setting (t + 1.75) equal to 4.069705149 and -4.069705149.Subproblem 1
t + 1.75 = 4.069705149 Simplifying t + 1.75 = 4.069705149 Reorder the terms: 1.75 + t = 4.069705149 Solving 1.75 + t = 4.069705149 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + t = 4.069705149 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + t = 4.069705149 + -1.75 t = 4.069705149 + -1.75 Combine like terms: 4.069705149 + -1.75 = 2.319705149 t = 2.319705149 Simplifying t = 2.319705149Subproblem 2
t + 1.75 = -4.069705149 Simplifying t + 1.75 = -4.069705149 Reorder the terms: 1.75 + t = -4.069705149 Solving 1.75 + t = -4.069705149 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + t = -4.069705149 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + t = -4.069705149 + -1.75 t = -4.069705149 + -1.75 Combine like terms: -4.069705149 + -1.75 = -5.819705149 t = -5.819705149 Simplifying t = -5.819705149Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.319705149, -5.819705149}Solution
t = {2.319705149, -5.819705149}
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