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Simplifying 15 = -16t2 + 34t + 5.7 Reorder the terms: 15 = 5.7 + 34t + -16t2 Solving 15 = 5.7 + 34t + -16t2 Solving for variable 't'. Combine like terms: 15 + -5.7 = 9.3 9.3 + -34t + 16t2 = 5.7 + 34t + -16t2 + -5.7 + -34t + 16t2 Reorder the terms: 9.3 + -34t + 16t2 = 5.7 + -5.7 + 34t + -34t + -16t2 + 16t2 Combine like terms: 5.7 + -5.7 = 0.0 9.3 + -34t + 16t2 = 0.0 + 34t + -34t + -16t2 + 16t2 9.3 + -34t + 16t2 = 34t + -34t + -16t2 + 16t2 Combine like terms: 34t + -34t = 0 9.3 + -34t + 16t2 = 0 + -16t2 + 16t2 9.3 + -34t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 9.3 + -34t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.58125 + -2.125t + t2 = 0 Move the constant term to the right: Add '-0.58125' to each side of the equation. 0.58125 + -2.125t + -0.58125 + t2 = 0 + -0.58125 Reorder the terms: 0.58125 + -0.58125 + -2.125t + t2 = 0 + -0.58125 Combine like terms: 0.58125 + -0.58125 = 0.00000 0.00000 + -2.125t + t2 = 0 + -0.58125 -2.125t + t2 = 0 + -0.58125 Combine like terms: 0 + -0.58125 = -0.58125 -2.125t + t2 = -0.58125 The t term is -2.125t. Take half its coefficient (-1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. -2.125t + 1.12890625 + t2 = -0.58125 + 1.12890625 Reorder the terms: 1.12890625 + -2.125t + t2 = -0.58125 + 1.12890625 Combine like terms: -0.58125 + 1.12890625 = 0.54765625 1.12890625 + -2.125t + t2 = 0.54765625 Factor a perfect square on the left side: (t + -1.0625)(t + -1.0625) = 0.54765625 Calculate the square root of the right side: 0.740038006 Break this problem into two subproblems by setting (t + -1.0625) equal to 0.740038006 and -0.740038006.Subproblem 1
t + -1.0625 = 0.740038006 Simplifying t + -1.0625 = 0.740038006 Reorder the terms: -1.0625 + t = 0.740038006 Solving -1.0625 + t = 0.740038006 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = 0.740038006 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = 0.740038006 + 1.0625 t = 0.740038006 + 1.0625 Combine like terms: 0.740038006 + 1.0625 = 1.802538006 t = 1.802538006 Simplifying t = 1.802538006Subproblem 2
t + -1.0625 = -0.740038006 Simplifying t + -1.0625 = -0.740038006 Reorder the terms: -1.0625 + t = -0.740038006 Solving -1.0625 + t = -0.740038006 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = -0.740038006 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = -0.740038006 + 1.0625 t = -0.740038006 + 1.0625 Combine like terms: -0.740038006 + 1.0625 = 0.322461994 t = 0.322461994 Simplifying t = 0.322461994Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.802538006, 0.322461994}
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