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Simplifying -4t2 + 108t + -136 = 0 Reorder the terms: -136 + 108t + -4t2 = 0 Solving -136 + 108t + -4t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(-34 + 27t + -1t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-34 + 27t + -1t2)' equal to zero and attempt to solve: Simplifying -34 + 27t + -1t2 = 0 Solving -34 + 27t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 34 + -27t + t2 = 0 Move the constant term to the right: Add '-34' to each side of the equation. 34 + -27t + -34 + t2 = 0 + -34 Reorder the terms: 34 + -34 + -27t + t2 = 0 + -34 Combine like terms: 34 + -34 = 0 0 + -27t + t2 = 0 + -34 -27t + t2 = 0 + -34 Combine like terms: 0 + -34 = -34 -27t + t2 = -34 The t term is -27t. Take half its coefficient (-13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. -27t + 182.25 + t2 = -34 + 182.25 Reorder the terms: 182.25 + -27t + t2 = -34 + 182.25 Combine like terms: -34 + 182.25 = 148.25 182.25 + -27t + t2 = 148.25 Factor a perfect square on the left side: (t + -13.5)(t + -13.5) = 148.25 Calculate the square root of the right side: 12.175795662 Break this problem into two subproblems by setting (t + -13.5) equal to 12.175795662 and -12.175795662.Subproblem 1
t + -13.5 = 12.175795662 Simplifying t + -13.5 = 12.175795662 Reorder the terms: -13.5 + t = 12.175795662 Solving -13.5 + t = 12.175795662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + t = 12.175795662 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + t = 12.175795662 + 13.5 t = 12.175795662 + 13.5 Combine like terms: 12.175795662 + 13.5 = 25.675795662 t = 25.675795662 Simplifying t = 25.675795662Subproblem 2
t + -13.5 = -12.175795662 Simplifying t + -13.5 = -12.175795662 Reorder the terms: -13.5 + t = -12.175795662 Solving -13.5 + t = -12.175795662 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + t = -12.175795662 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + t = -12.175795662 + 13.5 t = -12.175795662 + 13.5 Combine like terms: -12.175795662 + 13.5 = 1.324204338 t = 1.324204338 Simplifying t = 1.324204338Solution
The solution to the problem is based on the solutions from the subproblems. t = {25.675795662, 1.324204338}Solution
t = {25.675795662, 1.324204338}
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