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Simplifying 0 = -16t2 + 138t + 105 Reorder the terms: 0 = 105 + 138t + -16t2 Solving 0 = 105 + 138t + -16t2 Solving for variable 't'. Combine like terms: 0 + -105 = -105 -105 + -138t + 16t2 = 105 + 138t + -16t2 + -105 + -138t + 16t2 Reorder the terms: -105 + -138t + 16t2 = 105 + -105 + 138t + -138t + -16t2 + 16t2 Combine like terms: 105 + -105 = 0 -105 + -138t + 16t2 = 0 + 138t + -138t + -16t2 + 16t2 -105 + -138t + 16t2 = 138t + -138t + -16t2 + 16t2 Combine like terms: 138t + -138t = 0 -105 + -138t + 16t2 = 0 + -16t2 + 16t2 -105 + -138t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -105 + -138t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6.5625 + -8.625t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + -8.625t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + -8.625t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + -8.625t + t2 = 0 + 6.5625 -8.625t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 -8.625t + t2 = 6.5625 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 = 6.5625 + 18.59765625 Reorder the terms: 18.59765625 + -8.625t + t2 = 6.5625 + 18.59765625 Combine like terms: 6.5625 + 18.59765625 = 25.16015625 18.59765625 + -8.625t + t2 = 25.16015625 Factor a perfect square on the left side: (t + -4.3125)(t + -4.3125) = 25.16015625 Calculate the square root of the right side: 5.015990057 Break this problem into two subproblems by setting (t + -4.3125) equal to 5.015990057 and -5.015990057.Subproblem 1
t + -4.3125 = 5.015990057 Simplifying t + -4.3125 = 5.015990057 Reorder the terms: -4.3125 + t = 5.015990057 Solving -4.3125 + t = 5.015990057 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 = 5.015990057 + 4.3125 Combine like terms: -4.3125 + 4.3125 = 0.0000 0.0000 + t = 5.015990057 + 4.3125 t = 5.015990057 + 4.3125 Combine like terms: 5.015990057 + 4.3125 = 9.328490057 t = 9.328490057 Simplifying t = 9.328490057Subproblem 2
t + -4.3125 = -5.015990057 Simplifying t + -4.3125 = -5.015990057 Reorder the terms: -4.3125 + t = -5.015990057 Solving -4.3125 + t = -5.015990057 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 = -5.015990057 + 4.3125 Combine like terms: -4.3125 + 4.3125 = 0.0000 0.0000 + t = -5.015990057 + 4.3125 t = -5.015990057 + 4.3125 Combine like terms: -5.015990057 + 4.3125 = -0.703490057 t = -0.703490057 Simplifying t = -0.703490057Solution
The solution to the problem is based on the solutions from the subproblems. t = {9.328490057, -0.703490057}
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