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Simplifying 0 = 16t2 + 138t + 55 Reorder the terms: 0 = 55 + 138t + 16t2 Solving 0 = 55 + 138t + 16t2 Solving for variable 't'. Combine like terms: 0 + -55 = -55 -55 + -138t + -16t2 = 55 + 138t + 16t2 + -55 + -138t + -16t2 Reorder the terms: -55 + -138t + -16t2 = 55 + -55 + 138t + -138t + 16t2 + -16t2 Combine like terms: 55 + -55 = 0 -55 + -138t + -16t2 = 0 + 138t + -138t + 16t2 + -16t2 -55 + -138t + -16t2 = 138t + -138t + 16t2 + -16t2 Combine like terms: 138t + -138t = 0 -55 + -138t + -16t2 = 0 + 16t2 + -16t2 -55 + -138t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -55 + -138t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(55 + 138t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(55 + 138t + 16t2)' equal to zero and attempt to solve: Simplifying 55 + 138t + 16t2 = 0 Solving 55 + 138t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 3.4375 + 8.625t + t2 = 0 Move the constant term to the right: Add '-3.4375' to each side of the equation. 3.4375 + 8.625t + -3.4375 + t2 = 0 + -3.4375 Reorder the terms: 3.4375 + -3.4375 + 8.625t + t2 = 0 + -3.4375 Combine like terms: 3.4375 + -3.4375 = 0.0000 0.0000 + 8.625t + t2 = 0 + -3.4375 8.625t + t2 = 0 + -3.4375 Combine like terms: 0 + -3.4375 = -3.4375 8.625t + t2 = -3.4375 The t term is 8.625t. Take half its coefficient (4.3125). Square it (18.59765625) and add it to both sides. Add '18.59765625' to each side of the equation. 8.625t + 18.59765625 + t2 = -3.4375 + 18.59765625 Reorder the terms: 18.59765625 + 8.625t + t2 = -3.4375 + 18.59765625 Combine like terms: -3.4375 + 18.59765625 = 15.16015625 18.59765625 + 8.625t + t2 = 15.16015625 Factor a perfect square on the left side: (t + 4.3125)(t + 4.3125) = 15.16015625 Calculate the square root of the right side: 3.893604532 Break this problem into two subproblems by setting (t + 4.3125) equal to 3.893604532 and -3.893604532.Subproblem 1
t + 4.3125 = 3.893604532 Simplifying t + 4.3125 = 3.893604532 Reorder the terms: 4.3125 + t = 3.893604532 Solving 4.3125 + t = 3.893604532 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.3125' to each side of the equation. 4.3125 + -4.3125 + t = 3.893604532 + -4.3125 Combine like terms: 4.3125 + -4.3125 = 0.0000 0.0000 + t = 3.893604532 + -4.3125 t = 3.893604532 + -4.3125 Combine like terms: 3.893604532 + -4.3125 = -0.418895468 t = -0.418895468 Simplifying t = -0.418895468Subproblem 2
t + 4.3125 = -3.893604532 Simplifying t + 4.3125 = -3.893604532 Reorder the terms: 4.3125 + t = -3.893604532 Solving 4.3125 + t = -3.893604532 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-4.3125' to each side of the equation. 4.3125 + -4.3125 + t = -3.893604532 + -4.3125 Combine like terms: 4.3125 + -4.3125 = 0.0000 0.0000 + t = -3.893604532 + -4.3125 t = -3.893604532 + -4.3125 Combine like terms: -3.893604532 + -4.3125 = -8.206104532 t = -8.206104532 Simplifying t = -8.206104532Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.418895468, -8.206104532}Solution
t = {-0.418895468, -8.206104532}
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